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Application of Derivatives Set-1 MCQ Online Practice Test | Class 12 | Quiz Tweek

Application of Derivatives Set-1

Practice Important MCQs for Application of Derivatives — Mathematics, Class 12.

The chapter “Application of Derivatives” is crucial for both board exams and competitive tests because it directly develops tools for optimization, rate of change (Mean Value Theorem), tangent/approximation ideas (differentials), and inflection/extremum identification—skills that repeatedly appear in higher-grade questions.

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Q1. (Consider the function $f(x)=x^3-3x+1$ . Determine the intervals where $f$ is increasing and decreasing, and identify the local extrema (type and value).)

(A)

(Increasing on $(-1,1)$ ; decreasing on $(-\infty,-1)\cup(1,\infty)$ ; local minimum at $x=-1$ with $f(-1)=3$ , local maximum at $x=1$ with $f(1)=-1$ .)

(B)

(Increasing on $(-\infty,-1)\cup(1,\infty)$ ; decreasing on $(-1,1)$ ; local maximum at $x=1$ with $f(1)=-1$ , local minimum at $x=-1$ with $f(-1)=3$ .)

(C)

(Increasing on $(-\infty,-1)\cup(1,\infty)$ ; decreasing on $(-1,1)$ ; local maximum at $x=-1$ with $f(-1)=3$ , local minimum at $x=1$ with $f(1)=-1$ .)

(D)

(Increasing everywhere except at $x=\pm 1$ where $f'(x)=0$ ; no local extrema exist.)

Q2. (Find the point(s) on the parabola $y=x^2$ that are closest to the point $(0,3)$ .)

(A)

( $(0,0)$ )

(B)

( $\left(\pm\sqrt{\dfrac{5}{2}},\,\dfrac{5}{2}\right)$ )

(C)

( $(0,1)$ )

(D)

( $\left(\pm\sqrt{\dfrac{5}{4}},\,\dfrac{5}{4}\right)$ )

Q3. (Let $f(x)=\dfrac{x}{x+1}$ on $[1,3]$ . By the Mean Value Theorem find $c\in(1,3)$ such that $f'(c)=\dfrac{f(3)-f(1)}{3-1}$ .)

(A)

( $c=2\sqrt2-1$ )

(B)

( $c=2$ )

(C)

( $c=\sqrt2$ )

(D)

( $c=3-\sqrt2$ )

Q4. (Assertion (A): If $f$ is twice differentiable at $x_0$ and $f'(x_0)=0,\;f''(x_0)=0$ , then $x_0$ is necessarily a point of inflection of $f$ . Reason (R): A point of inflection is characterized by a change in sign of $f''$ at that point.)

(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, R is false.)

(D)

(A is false, R is true.)

Q5. (For real $k$ , determine the condition on $k$ such that the cubic equation $x^3-3kx+1=0$ has three distinct real roots.)

(A)

( $k>0$ )

(B)

( $k>2^{-2/3}$ )

(C)

( $0<k<2^{-2/3}$ )

(D)

( $k\ge 2^{-2/3}$ )

Q6. Using linear approximation (differentials), approximate the value of $\sqrt{4.05}$ by linearizing $f(x)=\sqrt{x}$ at $x=4$ .

(A)

$2.025$

(B)

$2.0125$

(C)

$2.005$

(D)

$2.020$

Q7. A rectangle is inscribed with its base on the $x$ -axis and its upper vertices on the parabola $y=12-x^2$ . Find the maximum possible area of such a rectangle.

(A)

$24$

(B)

$18$

(C)

$30$

(D)

$32$

Q8. For the function $f(x)=x^x$ defined for $x>0$ , the minimum value on $(0,\infty)$ is attained at which $x$ ?

(A)

$x=\dfrac{1}{e}$

(B)

$x=e$

(C)

$x=1$

(D)

$x=e^{-e}$

Q9. Assertion (A): Let $f$ be three times differentiable at $x=c$ . If $f'(c)=f''(c)=0$ and $f'''(c)\neq 0$ , then $x=c$ is a point of inflection of $f$ (and not a local extremum).
Reason (R): The Taylor expansion of $f$ about $c$ has the lowest-order nonzero term $\dfrac{f'''(c)}{6}(x-c)^3$ , an odd-powered term which changes sign across $c$ , causing a change of concavity.

(A)

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

(B)

A is true but R is false.

(C)

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

(D)

A is false but R is true.

Q10. Find the minimum value of $f(x)=\dfrac{x^2+4x+7}{x+1}$ for $x>-1$ .

(A)

Minimum value $5$

(B)

Minimum value $6$ (at $x=1$ )

(C)

Minimum value $7$

(D)

Minimum value $4$

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