10 Question MCQ (including Assertion) - Chapter 6 Class 12 Application of Derivatives

Question 1 of 10

Assertion (A): The function $f(x)=\sin (x)$ is increasing on the interval $\left(0, \frac{\pi}{2}\right)$.
Reason (R): The derivative, $f^{\prime}(x)=\cos (x)$, is positive for all $x$ in the interval ( $0, \frac{\pi}{2}$)

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.

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Question 2 of 10

Assertion (A): The function $f(x)=\log (\sin x)$ is strictly decreasing on $(\pi / 2, \pi)$.
Reason (R): The derivative $f^{\prime}(x)=\cot (x)$ is negative on the interval $(\pi / 2, \pi)$.

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.

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Question 3 of 10

NCERT Exemplar

A company's marginal cost is the derivative of its cost function, $C^{\prime}(x)$. If $C^{\prime}(100)=$ 15, what does this signify?

A The total cost of producing 100 items is $\$ 15$. B The average cost per item is $\$ 15$. C The cost to produce the 101st item is approximately $\$ 15$. D The company makes a profit of $\$ 15$ on the 100th item.

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Question 4 of 10

NCERT Exemplar

If $y=x^4-10$ and if $x$ changes from 2 to 1.99 , what is the change in $y$

A 0.32 B 0.032 C 5.68 D 5.968

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Question 5 of 10

NCERT Exemplar

The maximum value of $\sin x \cdot \cos x$ is

A $\frac{1}{4}$ B $\frac{1}{2}$ C $\sqrt{2}$ D $2 \sqrt{2}$

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Question 6 of 10

NCERT Exemplar

The function $f(x)=2 x^3-3 x^2-12 x+4$, has

A two points of local maximum B two points of local minimum C one maxima and one minima D no maxima or minima

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Question 7 of 10

NCERT Exemplar

For a function $f(x)$ on a closed interval $[a, b]$, the absolute maximum value can occur:

A Only where $f^{\prime}(x)=0$. B Only at the endpoints a or b . C Either at a point where $f^{\prime}(x)=0$ or at an endpoint. D Only where the function is increasing.

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Question 8 of 10

NCERT Exemplar

The function $f(x)=x^{\wedge} 3$ has $f^{\prime}(0)=0$. What is true about the point $x=0$ ?

A It is a local maximum. B It is a local minimum. C It is a point of inflection. D It is a point of discontinuity.

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Question 9 of 10

NCERT Exemplar

Finding the shortest distance from a point to a curve is an application of:

A The first derivative test for increasing/decreasing intervals. B Finding the absolute minimum of a distance function C The chain rule for related rates. D Finding points of inflection.

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Question 10 of 10

NCERT Exemplar

A function $f(x)$ represents the temperature of a chemical reaction over time. If $f^{\prime}(t)>\theta$ and $f^{\prime}(t)<0$ for a specific time interval, it means:

A The temperature is rising at an increasing rate. B The temperature is falling at a decreasing rate. C The temperature is rising, but the rate of increase is slowing down. D The temperature is falling at an increasing rate.

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10 Question MCQ (including Assertion) - Chapter 6 Class 12 Application of Derivatives