Class 12 Maths Chapter 5 Continuity and Differentiability – Exercise 5.5 NCERT Solutions

Introduction

Exercise 5.5 focuses on the Mean Value Theorem (MVT) and Rolle’s Theorem. Students learn how to verify conditions of these theorems, apply them to functions, and interpret their geometric meaning. This exercise is crucial for understanding the link between continuity, differentiability, and calculus applications.

Formulas Used

1. Rolle’s Theorem: If $f(x)$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then there exists $c\in (a,b)$ such that:

$$f^{\mathrm{\prime }}(c)=0$$

2. Mean Value Theorem (MVT): If $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c\in (a,b)$ such that:

$$f^{\mathrm{\prime }}(c)=\frac{f(b)-f(a)}{b-a}$$

Students Frequently Make Mistakes

• Forgetting to check continuity and differentiability conditions.

• Skipping verification of $f(a)=f(b)$ in Rolle’s Theorem.

• Misinterpreting geometric meaning of MVT.

• Errors in solving derivative equations for $c$.

• Confusing Rolle’s Theorem with MVT.

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

Q1. Verify Rolle’s Theorem for $f(x)=x^2-2x$ on $[0,2]$. $f(0)=0,f(2)=0$. Continuous and differentiable.

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

Q2. Verify Rolle’s Theorem for $f(x)=\sin x$ on $[0,\pi ]$. $f(0)=0,f(\pi )=0$. Continuous and differentiable.

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

Q3. Verify Rolle’s Theorem for $f(x)=x^3-3x$ on $[-\sqrt{3},\sqrt{3}]$. $f(-\sqrt{3})=f(\sqrt{3})=0$.

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

Q4. Verify MVT for $f(x)=x^2$ on $[1,3]$.

$$\frac{f(3)-f(1)}{3-1}=\frac{9-1}{2}=4$$

$$f^{\mathrm{\prime }}(x)=2x,f^{\mathrm{\prime }}(c)=4\Rightarrow c=2$$

Q5. Verify MVT for $f(x)=x^3$ on $[1,2]$.

$$\frac{f(2)-f(1)}{2-1}=7$$

$$f^{\mathrm{\prime }}(x)=3x^2,f^{\mathrm{\prime }}(c)=7\Rightarrow c=\sqrt{\frac{7}{3}}$$

Q6. Verify MVT for $f(x)=\sin x$ on $[0,\pi ]$.

$$\frac{f(\pi )-f(0)}{\pi -0}=\frac{0-0}{\pi }=0$$

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

Q7. Verify Rolle’s Theorem for $f(x)=\mathrm{\mid }x\mathrm{\mid }$ on $[-1,1]$. Not differentiable at $x=0$. Rolle’s Theorem not applicable.

Q8. Verify MVT for $f(x)=e^x$ on $[0,1]$.

$$\frac{f(1)-f(0)}{1-0}=e-1$$

$$f^{\mathrm{\prime }}(x)=e^x,f^{\mathrm{\prime }}(c)=e-1\Rightarrow c=\ln (e-1)$$

Q9. Verify Rolle’s Theorem for $f(x)=\cos x$ on $[0,2\pi ]$. $f(0)=f(2\pi )=1$. Continuous and differentiable.

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

Q10. Verify MVT for $f(x)=\ln x$ on $[1,e]$.

$$\frac{f(e)-f(1)}{e-1}=\frac{1-0}{e-1}=\frac{1}{e-1}$$

$$f^{\mathrm{\prime }}(x)=\frac{1}{x},f^{\mathrm{\prime }}(c)=\frac{1}{e-1}\Rightarrow c=e-1$$

FAQs (10)

FAQ1. What is Rolle’s Theorem? If $f(a)=f(b)$, then there exists $c$ with $f^{\mathrm{\prime }}(c)=0$.

FAQ2. What is MVT? There exists $c$ such that slope of tangent = slope of secant.

FAQ3. What conditions for Rolle’s Theorem? Continuity, differentiability, and $f(a)=f(b)$.

FAQ4. What conditions for MVT? Continuity and differentiability.

FAQ5. Does Rolle’s Theorem imply MVT? Yes, it is a special case of MVT.

FAQ6. What is geometric meaning of MVT? Tangent parallel to secant line.

FAQ7. Can discontinuous functions satisfy MVT? No.

FAQ8. Can non‑differentiable functions satisfy Rolle’s Theorem? No.

FAQ9. Why check conditions first? To ensure theorem applicability.

FAQ10. Why is Exercise 5.5 important? It builds foundation for calculus theorems and applications.

Conclusion

Exercise 5.5 has 10 solved questions and 10 FAQs that strengthen your understanding of Rolle’s Theorem and Mean Value Theorem. This completes the Continuity and Differentiability chapter in Class 12 Maths.

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