Class 12 Maths Chapter 6 Application of Derivatives – Exercise 6.2 NCERT Solutions

Introduction

Exercise 6.2 focuses on increasing and decreasing functions using derivatives. Students learn how to determine intervals where a function rises or falls by analyzing the sign of its derivative. This exercise is crucial for curve sketching, optimization, and calculus applications.

Formulas Used

1. Increasing Function: $f(x)$ is increasing on interval $I$ if:

$$f^{\mathrm{\prime }}(x)>0\mathrm{\forall }x\in I$$

2. Decreasing Function: $f(x)$ is decreasing on interval $I$ if:

$$f^{\mathrm{\prime }}(x)<0\mathrm{\forall }x\in I$$

3. Critical Points: Points where $f^{\mathrm{\prime }}(x)=0$ or undefined.

4. Test for Monotonicity:

   • Find derivative $f^{\mathrm{\prime }}(x)$.

   • Solve $f^{\mathrm{\prime }}(x)=0$ for critical points.

   • Check sign of $f^{\mathrm{\prime }}(x)$ in intervals divided by critical points.

Students Frequently Make Mistakes

• Forgetting to check derivative sign in each interval.

• Assuming derivative zero ⇒ maximum/minimum without testing intervals.

• Errors in solving inequalities for derivative sign.

• Confusing increasing with decreasing intervals.

• Skipping domain restrictions.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. Find intervals of monotonicity for $f(x)=x^2$.

$$f^{\mathrm{\prime }}(x)=2x$$

For $x<0$, $f^{\mathrm{\prime }}(x)<0$ ⇒ decreasing. For $x>0$, $f^{\mathrm{\prime }}(x)>0$ ⇒ increasing.

Q2. Find intervals of monotonicity for $f(x)=x^3$.

$$f^{\mathrm{\prime }}(x)=3x^2\ge 0$$

Always non‑negative ⇒ increasing everywhere.

Q3. Find intervals of monotonicity for $f(x)=x^2-4x+3$.

$$f^{\mathrm{\prime }}(x)=2x-4$$

Critical point: $x=2$. For $x<2$, $f^{\mathrm{\prime }}(x)<0$ ⇒ decreasing. For $x>2$, $f^{\mathrm{\prime }}(x)>0$ ⇒ increasing.

Q4. Find intervals of monotonicity for $f(x)=\sin x$ on $[0,2\pi ]$.

$$f^{\mathrm{\prime }}(x)=\cos x$$

Increasing on $[0,\pi \mathrm{/}2]\cup [3\pi \mathrm{/}2,2\pi ]$. Decreasing on $[\pi \mathrm{/}2,3\pi \mathrm{/}2]$.

Q5. Find intervals of monotonicity for $f(x)=\cos x$ on $[0,2\pi ]$.

$$f^{\mathrm{\prime }}(x)=-\sin x$$

Increasing on $[\pi ,2\pi ]$. Decreasing on $[0,\pi ]$.

Q6. Find intervals of monotonicity for $f(x)=e^x$.

$$f^{\mathrm{\prime }}(x)=e^x>0$$

Increasing everywhere.

Q7. Find intervals of monotonicity for $f(x)=\ln x$.

$$f^{\mathrm{\prime }}(x)=\frac{1}{x}>0,x>0$$

Increasing for all $x>0$.

Q8. Find intervals of monotonicity for $f(x)=x^4$.

$$f^{\mathrm{\prime }}(x)=4x^3$$

For $x<0$, $f^{\mathrm{\prime }}(x)<0$ ⇒ decreasing. For $x>0$, $f^{\mathrm{\prime }}(x)>0$ ⇒ increasing.

Q9. Find intervals of monotonicity for $f(x)=x^3-3x$.

$$f^{\mathrm{\prime }}(x)=3x^2-3=3(x^2-1)$$

Critical points: $x=\pm 1$. Decreasing on $(-1,1)$. Increasing on $(-\mathrm{\infty },-1)\cup (1,\mathrm{\infty })$.

Q10. Find intervals of monotonicity for $f(x)=\tan x$ on $(-\pi \mathrm{/}2,\pi \mathrm{/}2)$.

$$f^{\mathrm{\prime }}(x)={\sec }^{2}x>0$$

Increasing everywhere in domain.

FAQs (10)

FAQ1. What is increasing function? Function with positive derivative in interval.

FAQ2. What is decreasing function? Function with negative derivative in interval.

FAQ3. What is critical point? Point where derivative = 0 or undefined.

FAQ4. How to test monotonicity? Check sign of derivative in intervals.

FAQ5. Is $x^3$ increasing everywhere? Yes.

FAQ6. Is $x^2$ increasing everywhere? No, decreasing for $x<0$.

FAQ7. Is exponential function increasing everywhere? Yes.

FAQ8. Is logarithmic function increasing everywhere? Yes, for $x>0$.

FAQ9. Can trigonometric functions be monotonic? Yes, in restricted intervals.

FAQ10. Why is Exercise 6.2 important? It builds foundation for curve sketching and optimization.

Conclusion

Exercise 6.2 has 10 solved questions and 10 FAQs that strengthen your understanding of increasing and decreasing functions using derivatives. This builds the foundation for optimization and curve analysis in Class 12 Maths.

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