Application of Derivative (Maths) Question Bank - Engineer & Medical

Q1.

Consider the function $f: (0, \infty) \rightarrow (-\infty, \infty)$ given by

$f(x) = \sqrt{x}\, \log_e(x) - x + 1$.

Then which one of the following statements is TRUE?

(A)

The derivative of the function $f$ is decreasing in the interval $(0, 1)$

(B)

The function $f$ has a local maximum at some point $a \in (0, \infty)$

(C)

The function $f$ has a local minimum at some point $b \in (0, \infty)$

(D)

The function $f$ has NEITHER a point of local maximum NOR a point of local minimum in the interval $(0, \infty)$

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Q2.

Let $\mathbb{R}$ denote the set of all real numbers. For a real number $x$, let $[x]$ denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number. Match each entry in List-I to the correct entry in List-II and choose the correct option.

	LIST - I		LIST - II
(P)	The minimum value of $n$ for which the function $f(x)=\left[\frac{10 x^3-45 x^2+60 x+35}{n}\right]$ is continuous on the interval $[1,2]$,is	(A)	8
(Q)	The minimum value of $n$ for which $g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right),x \in \mathbb{R}$,is an increasing function on $\mathbb{R}$,is	(B)	9
(R)	The smallest natural number $n$ which is greater than 5,such that $x=3$ is a point of local minima of $h(x)=\left(x^2-9\right)^{\mathrm{n}}\left(x^2+2 x+3\right)$,is	(C)	5
(S)	Number of $x_0 \in \mathbb{R}$ such that $l(x)=\sum_{k=0}^4\left(\sin \|x-k\|+\cos \left\|x-k+\frac{1}{2}\right\|\right),x \in \mathbb{R}$,is NOT differentiable at $x_0$,is	(D)	6
		(5)	10

(A)

$(\mathrm{P}) \rightarrow(1),(\mathrm{Q}) \rightarrow(3),(\mathrm{R}) \rightarrow(2),(\mathrm{S}) \rightarrow(5)$

(B)

$(\mathrm{P}) \rightarrow(2),(\mathrm{Q}) \rightarrow(1),(\mathrm{R}) \rightarrow(4),(\mathrm{S}) \rightarrow(3)$

(C)

$(\mathrm{P}) \rightarrow(5),(\mathrm{Q}) \rightarrow(1),(\mathrm{R}) \rightarrow(4),(\mathrm{S}) \rightarrow(3)$

(D)

$(\mathrm{P}) \rightarrow(2),(\mathrm{Q}) \rightarrow(3),(\mathrm{R}) \rightarrow(1),(\mathrm{S}) \rightarrow(5)$

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