Application of Derivatives – Increasing & Decreasing Functions
Application of Derivatives mein derivative ka use sirf slope find karne ke liye nahi hota. Derivative ka ek very important application hai kisi function ka behaviour study karna — yani function kis interval mein increase kar raha hai aur kis interval mein decrease.
JEE Main mein is concept se direct questions aa sakte hain, jaise:
- Function ke increasing intervals find karo.
- Function ke decreasing intervals find karo.
- Critical points find karo.
- Local maximum/minimum identify karo.
- Parameter ki values find karo jinke liye function all real numbers par increasing/decreasing ho.
Is topic ka core idea bahut simple hai:
f′(x) > 0 → Increasing
f′(x) < 0 → Decreasing
Aur isi idea ko sign chart ke saath use karke JEE Main ke majority increasing-decreasing questions systematically solve kiye ja sakte hain.
1. Basic Idea of Increasing & Decreasing Functions
Sabse pehle simple language mein samjho ki increasing aur decreasing function ka actual meaning kya hai.
Increasing Function
Agar x ki value increase karne par function ki value bhi increase kare, toh function increasing hai.
x ↑ → f(x) ↑
Derivative ke through:
f′(x) > 0 → Function is strictly increasing
Decreasing Function
Agar x ki value increase karne par function ki value decrease kare, toh function decreasing hai.
x ↑ → f(x) ↓
Derivative test:
f′(x) < 0 → Function is strictly decreasing
Isliye ek line yaad rakho:
Positive Slope → Increasing
Negative Slope → Decreasing
2. Increasing Function – Example
Consider:
f(x) = x²
Differentiate:
f′(x) = 2x
Function increasing hone ke liye:
2x > 0
Therefore:
x > 0
Hence x² is increasing on:
(0, ∞)
Graphically bhi yeh samajh sakte ho: x = 0 ke right side par parabola upward ja raha hai, isliye function increase kar raha hai.
3. Decreasing Function – Example
Same function:
f(x) = x²
Derivative:
f′(x) = 2x
Decreasing hone ke liye:
2x < 0
Therefore:
x < 0
Hence x² is decreasing on:
(−∞, 0)
So complete result:
x² is decreasing on (−∞, 0)
x² is increasing on (0, ∞)
Yeh example increasing-decreasing functions ka basic structure samajhne ke liye bahut important hai.
4. Critical Points
Increasing/decreasing intervals find karte waqt humein sabse pehle critical points identify karne hote hain.
Basic procedure:
f′(x) = 0
solve karo.
Saath hi un points ko bhi check karna hota hai jahan:
f′(x) does not exist
lekin function khud defined ho.
Example
Given:
f(x) = x² − 4x + 3
Differentiate:
f′(x) = 2x − 4
Critical point ke liye:
2x − 4 = 0
Therefore:
x = 2
Hence x = 2 is a critical point.
Important: Sirf f′(x)=0 milne se automatically maximum ya minimum prove nahi hota. Uske liye derivative sign change check karna useful hai.
5. Sign Chart Method – Most Important JEE Method
JEE Main mein increasing/decreasing questions solve karne ka sabse systematic method hai sign chart method.
STEP 1 – Find f′(x)
Given function ko differentiate karo.
STEP 2 – Solve f′(x) = 0
Jo values milti hain woh important critical points hain.
STEP 3 – Critical Points Arrange Karo
Number line par critical points ko increasing order mein arrange karo.
STEP 4 – Intervals Banao
Critical points number line ko multiple intervals mein divide karenge.
STEP 5 – Har Interval Mein f′(x) Ka Sign Check Karo
Har interval se ek convenient test point lekar derivative ka sign determine karo.
STEP 6 – Increasing/Decreasing Identify Karo
+ → Increasing
− → Decreasing
6. Sign Chart Example
Consider:
f′(x) = (x − 1)(x + 2)
Step 1 – Critical Points
Put:
(x − 1)(x + 2) = 0
Therefore:
x = 1, −2
Step 2 – Intervals
Critical points are:
−2 and 1
Therefore intervals:
- (−∞, −2)
- (−2, 1)
- (1, ∞)
Step 3 – Sign Check
| Interval | Sign of f′(x) | Behaviour |
|---|---|---|
| (−∞, −2) | + | Increasing |
| (−2, 1) | − | Decreasing |
| (1, ∞) | + | Increasing |
Therefore:
Increasing: (−∞, −2) ∪ (1, ∞)
Decreasing: (−2, 1)
7. Local Maximum and Minimum Connection
Derivative ka sign change sirf increasing/decreasing intervals batane ke liye nahi, balki local maxima aur minima identify karne ke liye bhi useful hai.
Local Maximum
Agar derivative ka sign:
+ → −
ho jaye, toh function:
Increasing → Decreasing
Isliye point par Local Maximum hota hai.
Local Minimum
Agar derivative ka sign:
− → +
ho jaye, toh function:
Decreasing → Increasing
Isliye point par Local Minimum hota hai.
No Sign Change
Agar sign:
+ → +
ya:
− → −
ho, toh generally local maximum/minimum nahi hota.
| Derivative Sign Change | Result |
|---|---|
| + → − | Local Maximum |
| − → + | Local Minimum |
| + → + | No local extremum generally |
| − → − | No local extremum generally |
8. Important Conditions – Strict vs Non-Strict
JEE questions mein wording ko carefully read karna bahut important hai.
Strictly Increasing
Basic derivative test:
f′(x) > 0
Toh function strictly increasing hota hai on the interval.
Non-Decreasing
Agar:
f′(x) ≥ 0
toh function non-decreasing behaviour show kar sakta hai.
Strictly Decreasing
f′(x) < 0
Non-Increasing
f′(x) ≤ 0
Important nuance: derivative ka zero hona by itself function ko non-increasing ya non-decreasing nahi banata. Sign behaviour poore interval par consider karna hota hai.
Isi wajah se isolated points par f′(x)=0 hone ke bawajood function increasing reh sakta hai.
9. Parameter-Based Questions
JEE Main mein ek very important pattern hota hai:
"Find the values of parameter a for which the function is increasing for all real x."
Aise questions mein direct function ko increasing bolne ke bajay derivative ki condition lagani hoti hai.
Example
Find the values of a for which:
f(x) = x³ − ax² + 3x
is increasing for all real x.
Step 1 – Differentiate
f′(x) = 3x² − 2ax + 3
Step 2 – Increasing Condition
All real x ke liye increasing/non-decreasing condition ke liye:
f′(x) ≥ 0
Therefore:
3x² − 2ax + 3 ≥ 0
Step 3 – Quadratic Condition
Quadratic:
3x² − 2ax + 3
Har real x ke liye non-negative hone ke liye:
Discriminant ≤ 0
Therefore:
(−2a)² − 4(3)(3) ≤ 0
4a² − 36 ≤ 0
a² ≤ 9
Therefore:
−3 ≤ a ≤ 3
Yeh JEE Main ka very important parameter pattern hai.
10. JEE Shortcut – Quadratic Derivative
Suppose derivative quadratic form mein hai:
f′(x) = ax² + bx + c
For Increasing for All Real x
Humein generally:
f′(x) ≥ 0
chahiye.
Iske liye:
- a > 0
- D = b² − 4ac ≤ 0
For Decreasing for All Real x
Humein:
f′(x) ≤ 0
chahiye.
Is case mein:
- a < 0
- D = b² − 4ac ≤ 0
Quick memory trick:
Increasing → Upward Quadratic + No Sign Crossing
Decreasing → Downward Quadratic + No Sign Crossing
11. JEE Main-Level Example – Find Increasing Intervals
Question: Find the intervals where:
f(x) = x³ − 3x² − 9x + 5
is increasing.
Step 1 – Differentiate
f′(x) = 3x² − 6x − 9
Factorise:
f′(x) = 3(x² − 2x − 3)
f′(x) = 3(x − 3)(x + 1)
Step 2 – Critical Points
x = 3, −1
Step 3 – Sign Chart
| Interval | Sign of f′(x) | Behaviour |
|---|---|---|
| (−∞, −1) | + | Increasing |
| (−1, 3) | − | Decreasing |
| (3, ∞) | + | Increasing |
Final Answer
Therefore the function is increasing on:
(−∞, −1) ∪ (3, ∞)
And decreasing on:
(−1, 3)
12. Understanding Why Sign Chart Works
Sign chart ko sirf procedure ki tarah yaad mat karo. Iska mathematical meaning bhi samjho.
Derivative essentially function ka local slope batata hai.
- Positive derivative → graph ka slope positive → graph upward trend mein.
- Negative derivative → graph ka slope negative → graph downward trend mein.
- Zero derivative → tangent horizontal ho sakti hai.
Isi liye derivative ka sign directly function ke increasing/decreasing behaviour se connected hai.
Derivative = Slope → Slope Sign = Function Behaviour
13. Critical Point ≠ Always Maximum or Minimum
JEE mein ek common trap hai:
f′(x) = 0 → Maximum/Minimum
Yeh statement automatically true nahi hai.
Example:
f(x) = x³
Derivative:
f′(x) = 3x²
At x = 0:
f′(0) = 0
Lekin derivative ka sign:
+ → +
hai. Isliye x = 0 par local maximum/minimum nahi hota.
Is example se important lesson:
Critical Point milna ≠ Extremum confirm hona
14. Common JEE Mistakes
Is topic mein students mostly concept se nahi, interpretation aur sign analysis se mistakes karte hain.
- f′(x)=0 ko directly maximum/minimum maan lena.
- Critical points find karke sign test skip kar dena.
- Increasing aur decreasing intervals reverse likh dena.
- "For all real x" condition ignore karna.
- Parameter questions mein discriminant condition bhool jana.
- Strictly increasing aur non-decreasing ko same samajhna.
- Derivative undefined points ko check na karna jab function defined ho.
- Final intervals ko critical points ke around properly divide na karna.
Golden Rule
Derivative → Critical Points → Sign Chart → Interval
15. JEE Quick-Solving Strategy
Exam hall mein increasing/decreasing function ka question dekhte hi yeh fixed approach follow karo:
- Function differentiate karo.
- f′(x)=0 solve karo.
- Derivative undefined points check karo, if applicable.
- Critical points number line par arrange karo.
- Intervals identify karo.
- Har interval mein derivative ka sign determine karo.
- + → Increasing.
- − → Decreasing.
- +→− → Local Maximum.
- −→+ → Local Minimum.
Differentiate → Solve → Divide → Sign Check → Answer
16. Complete Formula Sheet
| Concept | Condition / Formula |
|---|---|
| Increasing | f′(x) > 0 |
| Decreasing | f′(x) < 0 |
| Non-decreasing | f′(x) ≥ 0 |
| Non-increasing | f′(x) ≤ 0 |
| Critical Point | f′(x)=0 or derivative does not exist while function exists |
| Local Maximum | f′: + → − |
| Local Minimum | f′: − → + |
| Increasing for all real x, quadratic derivative | a > 0, D ≤ 0 |
| Decreasing for all real x, quadratic derivative | a < 0, D ≤ 0 |
17. One-Minute Revision
Exam se just pehle sirf yeh points revise karo:
f′(x) > 0 → Increasing
f′(x) < 0 → Decreasing
f′(x)=0 or DNE → Critical Point candidate
+ → − → Maximum
− → + → Minimum
+ → + / − → − → No local extremum generally
For all real x → Check derivative sign for every real x
Derivative → Critical Points → Sign Chart → Increasing/Decreasing
18. Final Revision Box
Main Test
f′(x) > 0 → Increasing
f′(x) < 0 → Decreasing
Critical Point
f′(x)=0 or derivative does not exist while the function exists.
Extrema
+ → − → Local Maximum
− → + → Local Minimum
Parameter Shortcut
For f′(x)=ax²+bx+c:
Increasing for all real x → a > 0 and D ≤ 0
Decreasing for all real x → a < 0 and D ≤ 0
Golden Method
Differentiate → Find Critical Points → Make Sign Chart → Identify Intervals
```19. Practice Questions
Question 1
Find the intervals where:
f(x)=x³−3x
is increasing.
Hint: Find f′(x), solve f′(x)=0 and make a sign chart.
Answer: (−∞, −1) ∪ (1, ∞)
Question 2
Find the intervals where:
f(x)=−x²+4x+1
is increasing and decreasing.
Answer: Increasing on (−∞, 2) and decreasing on (2, ∞).
Question 3
If:
f′(x)=(x−4)(x+2)
identify the increasing and decreasing intervals.
Answer: Increasing on (−∞,−2) ∪ (4,∞); decreasing on (−2,4).
Question 4
Find the values of a for which:
f(x)=x³−ax²+3x
is increasing for all real x.
Answer: −3 ≤ a ≤ 3
20. PDF Notes – Application of Derivatives
Increasing and decreasing functions ke detailed board notes ko PDF format mein revise karne ke liye neeche PDF embed ki gayi hai.
21. Frequently Asked Questions (FAQs)
Q1. Increasing function ke liye derivative ka sign kya hota hai?
Strictly increasing behaviour ke liye basic test: f′(x) > 0.
Q2. Decreasing function ke liye derivative ka sign kya hota hai?
Strictly decreasing behaviour ke liye: f′(x) < 0.
Q3. Critical point kya hota hai?
Aisa point jahan f′(x)=0 ya derivative exist nahi karta, jabki function us point par defined ho.
Q4. Kya f′(x)=0 ka matlab maximum ya minimum hota hai?
Nahi. f′(x)=0 sirf critical point candidate deta hai. Maximum/minimum confirm karne ke liye derivative sign change check karna useful hai.
Q5. Local maximum ka derivative sign change kya hota hai?
+ → −, yani function increasing se decreasing ho jata hai.
Q6. Local minimum ka derivative sign change kya hota hai?
− → +, yani function decreasing se increasing ho jata hai.
Q7. Increasing for all real x question mein kya karna hai?
Pehle derivative find karo aur ensure karo ki derivative required non-negative condition ko all real x ke liye satisfy kare. Agar derivative quadratic hai, toh leading coefficient aur discriminant conditions useful hoti hain.
Q8. Quadratic derivative all real x par non-negative kab hota hai?
ax²+bx+c ≥ 0 for all real x ke liye, basic non-degenerate quadratic case mein a > 0 aur D ≤ 0 required hai.
Q9. Sign chart kyun banate hain?
Critical points ke beech derivative ka sign determine karne ke liye sign chart banaya jata hai. Isse increasing aur decreasing intervals directly identify ho jate hain.
Q10. JEE Main mein fastest method kya hai?
Differentiate → Critical Points → Sign Chart → Answer
Final Thoughts
Application of Derivatives – Increasing & Decreasing Functions ka core concept actually very simple hai. Humein bas derivative ke sign ko function ke behaviour se connect karna hai.
Positive Derivative → Increasing
Negative Derivative → Decreasing
JEE Main questions ke liye sabse important workflow hai:
Differentiate → Find Critical Points → Make Sign Chart → Identify Intervals
Aur agar question mein parameter diya ho aur function ko all real x ke liye increasing/decreasing banana ho, toh derivative ki sign condition ko poore real number system par apply karo. Quadratic derivative ke case mein discriminant shortcut especially useful hai.
Ek final line yaad rakho:
Derivative tells you the slope, and the sign of the slope tells you the behaviour of the function.