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Continuity and Differentiability Set-4 MCQ Online Practice Test | Class 12 | Quiz Tweek

Continuity and Differentiability Set-4

Practice Important MCQs for Continuity and Differentiability — Mathematics, Class 12.

The chapter “Continuity and Differentiability” is central in Class 12 Mathematics because it links limits, continuity, and tangent-slope behavior of functions. Board questions typically test continuity/differentiability at a point using limits and derivatives, while competitive exams often ask conceptual understanding via inverse functions, mean value theorem ideas, and carefully constructed piecewise functions.

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Q1. Let

f(x)=\begin{aligned} x^2\sin\left(\frac{1}{x}\right) &\text{ if } x\ne 0 <br> 0 &\text{ if } x=0 \end{aligned}

. Then $f'(0)=$ ?

(A)

$1$

(B)

$0$

(C)

$f'(0)$ does not exist

(D)

$\dfrac{1}{2}$

Q2. Let

f(x)=\begin{aligned} \dfrac{e^{ax}-1}{x} &\text{ if } x<0 <br> \ln(1+bx) &\text{ if } x\ge 0 \end{aligned}

. For which real numbers $a,b$ is $f$ differentiable at $x=0$ ?

(A)

$a=b=1$

(B)

$a=1,\;b=0$

(C)

$a=0,\;b=1$

(D)

$a=b=0$

Q3. Let $f$ be continuous on $[0,1]$ and differentiable on $(0,1)$ with $f(0)=f(1)=0$ and $|f'(x)|\le M$ for all $x\in(0,1)$ . The best possible upper bound for $\max_{x\in[0,1]}|f(x)|$ is:

(A)

$\dfrac{M}{2}$

(B)

$M$

(C)

$\dfrac{M}{4}$

(D)

$\dfrac{M}{3}$

Q4. Consider

f(x)=\begin{aligned} x^2\sin\left(\frac{1}{x}\right) &\text{ if } x\ne0 <br> 0 &\text{ if } x=0 \end{aligned},\qquad g(x)=\begin{aligned} x\sin\left(\frac{1}{x^2}\right) &\text{ if } x\ne0 <br> 0 &\text{ if } x=0 \end{aligned}

. Which statement about differentiability at $x=0$ is correct?

(A)

Both $f$ and $g$ are differentiable at $0$

(B)

Neither $f$ nor $g$ is differentiable at $0$

(C)

$f$ is differentiable at $0$ but $g$ is not

(D)

$f$ is not differentiable at $0$ but $g$ is

Q5. Does there exist a differentiable function $f:\mathbb{R}\to\mathbb{R}$ such that

f'(x)=\begin{aligned} 1 &\text{ if } x\ne0 <br> 0 &\text{ if } x=0 \end{aligned}

? Choose the correct statement.

(A)

Yes, such functions exist

(B)

No, no such function exists

(C)

Yes, such a function exists but $f'$ must be discontinuous at $0$

(D)

Such a function exists only if $f(0)=0$

Q6. Let $f(x)=x^2\sin\frac{1}{x}$ for $x\ne0$ and $f(0)=0$ . Then $f'(0)=$

(A)

$-1$

(B)

$1$

(C)

$0$

(D)

does not exist

Q7. Let $f(x)=kx^3+3x$ for $x\le 1$ and $f(x)=x^3+kx+2$ for $x>1$ . For which value of $k$ is $f$ differentiable at $x=1$ ?

(A)

$k=1$

(B)

$k=0$

(C)

$k=-1$

(D)

$k=\dfrac{3}{2}$

Q8. Let $f(x)=x^3+x$ and let $g$ be the inverse function of $f$ . Then $g'(2)=$

(A)

$\dfrac{1}{3}$

(B)

$\dfrac{1}{2}$

(C)

$\dfrac{1}{7}$

(D)

$\dfrac{1}{4}$

Q9. For real $\alpha$ , define $f_\alpha(x)=|x|^\alpha\sin\frac{1}{x}$ for $x\ne0$ and $f_\alpha(0)=0$ . For which values of $\alpha$ is $f_\alpha$ differentiable at $0$ ?

(A)

$\alpha>1$

(B)

$\alpha\ge 1$

(C)

$\alpha>0$

(D)

$\alpha\le 1$

Q10. Let $f$ be continuous on $\mathbb{R}$ and differentiable on $\mathbb{R}\setminus\{0\}$ . Suppose $\displaystyle\lim_{x\to0}f'(x)=L$ exists (finite). Which conclusion is valid?

(A)

$f$ may fail to be differentiable at $0$ even if $\lim_{x\to0}f'(x)$ exists

(B)

$f$ is differentiable at $0$ and $f'(0)=L$

(C)

$f$ is differentiable at $0$ but $f'(0)$ need not equal $L$

(D)

the limit $L$ must be $0$

Q11. Let $f(x)=x^2\sin\frac{1}{x}$ for $x\ne0$ and $f(0)=0$ . Which of the following is true about $f$ at $x=0$ ?

(A)

$f$ is not continuous at $0$ .

(B)

$f$ is continuous at $0$ but not differentiable at $0$ .

(C)

$f$ is differentiable at $0$ and $f'(0)=0$ .

(D)

$f$ is differentiable at $0$ and $f'(0)=1$ .

Q12. Find real numbers $a,b$ such that the function $f$ defined by $f(x)=ax+b$ for $x\le1$ and $f(x)=x^2+1$ for $x>1$ is differentiable at $x=1$ .

(A)

$(a,b)=(1,1)$

(B)

$(a,b)=(2,1)$

(C)

$(a,b)=(0,2)$

(D)

$(a,b)=(2,0)$

Q13. Let $f(x)=\dfrac{x}{1+x^2}$ for all real $x$ . Which of the following statements is correct?

(A)

$f$ is decreasing on $(0,1)$ .

(B)

$f$ is increasing on $(-1,1)$ and attains its global maximum $\tfrac{1}{2}$ at $x=1$ .

(C)

$f$ has no stationary points.

(D)

$f$ is even and attains its minimum $-\tfrac{1}{2}$ at $x=1$ .

Q14. Assertion (A): If $f$ is differentiable on an interval $I$ , one-to-one on $I$ , and $f'(x)\ne0$ for all $x\in I$ , then the inverse function $f^{-1}$ is differentiable on $f(I)$ .
Reason (R): For $y\in f(I)$ , $(f^{-1})'(y)=\dfrac{1}{f'(f^{-1}(y))}$ .

(A)

Both A and R are true and R is the correct explanation of A.

(B)

Both A and R are true but R is not the correct explanation of A.

(C)

A is true but R is false.

(D)

A is false but R is true.

Q15. Let $f(x)=x^2\sin\frac{1}{x}$ and $g(x)=x^3\sin\frac{1}{x}$ for $x\ne0$ , and $f(0)=g(0)=0$ . Which of the following statements about differentiability and continuity of derivatives at $0$ is correct?

(A)

Both $f$ and $g$ are differentiable at $0$ and both $f'$ and $g'$ are continuous at $0$ .

(B)

Neither $f$ nor $g$ is differentiable at $0$ .

(C)

Both $f$ and $g$ are differentiable at $0$ ; $f'$ is not continuous at $0$ whereas $g'$ is continuous at $0$ .

(D)

$f$ is not differentiable at $0$ but $g$ is differentiable at $0$ and $g'$ is not continuous at $0$ .

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