Class 12 Maths Chapter 6 Application of Derivatives – Exercise 6.3 NCERT Solutions

Introduction

Exercise 6.3 focuses on maxima and minima of functions using derivatives. Students learn how to find turning points, classify them as maxima or minima, and solve optimization problems. This exercise is essential for curve sketching, economics, and real‑life applications of calculus.

Formulas Used

1. Critical Points:

$$f^{\mathrm{\prime }}(x)=0\text{or}{f}^{\mathrm{\prime }}(x)\text{ undefined}$$

2. Second Derivative Test:

   • If $f^{\mathrm{\prime }\mathrm{\prime }}(x)>0$ at critical point ⇒ local minimum.

   • If $f^{\mathrm{\prime }\mathrm{\prime }}(x)<0$ at critical point ⇒ local maximum.

   • If $f^{\mathrm{\prime }\mathrm{\prime }}(x)=0$ ⇒ test inconclusive.

3. Optimization Problems: Use derivative to maximize or minimize given quantity.

Students Frequently Make Mistakes

• Forgetting to check second derivative for classification.

• Assuming $f^{\mathrm{\prime }}(x)=0$ always gives maxima/minima.

• Errors in solving derivative equations.

• Confusing local maxima/minima with global maxima/minima.

• Skipping domain restrictions in optimization problems.

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

Q1. Find maxima/minima of $f(x)=x^2$.

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

$$f^{\mathrm{\prime }\mathrm{\prime }}(x)=2>0\Rightarrow \text{minimum at }x=0$$

Q2. Find maxima/minima of $f(x)=-x^2$.

$$f^{\mathrm{\prime }}(x)=-2x=0\Rightarrow x=0$$

$$f^{\mathrm{\prime }\mathrm{\prime }}(x)=-2<0\Rightarrow \text{maximum at }x=0$$

Q3. Find maxima/minima of $f(x)=x^3$.

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

$$f^{\mathrm{\prime }\mathrm{\prime }}(x)=6x=0\Rightarrow \text{test inconclusive}$$

No maxima/minima, only point of inflection at $x=0$.

Q4. Find maxima/minima of $f(x)=x^3-3x$.

$$f^{\mathrm{\prime }}(x)=3x^2-3=0\Rightarrow x=\pm 1$$

$$f^{\mathrm{\prime }\mathrm{\prime }}(x)=6x$$

At $x=1,f^{\mathrm{\prime }\mathrm{\prime }}(1)=6>0$ ⇒ local minimum. At $x=-1,f^{\mathrm{\prime }\mathrm{\prime }}(-1)=-6<0$ ⇒ local maximum.

Q5. Find maxima/minima of $f(x)=x^4$.

$$f^{\mathrm{\prime }}(x)=4x^3=0\Rightarrow x=0$$

$$f^{\mathrm{\prime }\mathrm{\prime }}(x)=12x^2=0\Rightarrow \text{test inconclusive}$$

Minimum at $x=0$ by inspection.

Q6. Find maxima/minima of $f(x)=x^2+4x+5$.

$$f^{\mathrm{\prime }}(x)=2x+4=0\Rightarrow x=-2$$

$$f^{\mathrm{\prime }\mathrm{\prime }}(x)=2>0\Rightarrow \text{minimum at }x=-2$$

Q7. Find maxima/minima of $f(x)=x^2-4x+3$.

$$f^{\mathrm{\prime }}(x)=2x-4=0\Rightarrow x=2$$

$$f^{\mathrm{\prime }\mathrm{\prime }}(x)=2>0\Rightarrow \text{minimum at }x=2$$

Q8. Find maxima/minima of $f(x)=\sin x$ on $[0,2\pi ]$.

$$f^{\mathrm{\prime }}(x)=\cos x=0\Rightarrow x=\frac{\pi }{2},\frac{3\pi }{2}$$

At $x=\pi \mathrm{/}2,f^{\mathrm{\prime }\mathrm{\prime }}(x)=-\sin x=-1<0$ ⇒ maximum. At $x=3\pi \mathrm{/}2,f^{\mathrm{\prime }\mathrm{\prime }}(x)=1>0$ ⇒ minimum.

Q9. Find maxima/minima of $f(x)=\cos x$ on $[0,2\pi ]$.

$$f^{\mathrm{\prime }}(x)=-\sin x=0\Rightarrow x=0,\pi ,2\pi$$

At $x=0,2\pi$ ⇒ maxima. At $x=\pi$ ⇒ minimum.

Q10. Solve optimization: Find dimensions of rectangle of perimeter 20 units with maximum area. Let sides be $x,y$.

$$2(x+y)=20\Rightarrow y=10-x$$

Area $A=xy=x(10-x)=10x-x^2$.

$$A^{\mathrm{\prime }}(x)=10-2x=0\Rightarrow x=5$$

$$y=5,\text{maximum area at square }5\times 5$$

FAQs (10)

FAQ1. What is maxima? Point where function attains highest value locally.

FAQ2. What is minima? Point where function attains lowest value locally.

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

FAQ4. What is second derivative test? Method to classify maxima/minima.

FAQ5. What is point of inflection? Point where curve changes concavity.

FAQ6. Can maxima/minima exist without derivative? Yes, at endpoints or undefined points.

FAQ7. Is $x^3$ having maxima/minima? No, only inflection at 0.

FAQ8. Is quadratic function always having maxima/minima? Yes, depending on sign of coefficient.

FAQ9. Why optimization important? Used in maximizing/minimizing real‑life quantities.

FAQ10. Why is Exercise 6.3 important? It builds foundation for maxima/minima and optimization problems.

Conclusion

Exercise 6.3 has 10 solved questions and 10 FAQs that strengthen your understanding of maxima, minima, and optimization using derivatives. This builds the foundation for advanced calculus applications in Class 12 Maths.

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