Statement - 1: The function x^2 (e^x + e^ -x ) is increasing for … - Vidyadip

MCQ

Statement $- 1:$ The function $x^2 (e^x + e^{-x})$ is increasing for all $x > 0.$

Statement $-2:$ The functions $x^2e^x$ and $x^2e^{-x}$ are increasing for all $x > 0$ and the sum of two increasing functions in any interval $(a, b)$ is an increasing function in $(a, b).$

A
Statement $-1$ is false; Statement $-2$ is true.
B
Statement $-1$ is true; Statement $- 2$ is true;Statement $-2$ is not a correct explanation for Statement $- 1.$
✓
Statement $-1$ is true; Statement $-2$ is false.
D
Statement $- 1$ is true; Statement $-2$ is true; Statement $-2$ is a correct explanation for statement $-1.$

✓

Answer

Correct option: C.

Statement $-1$ is true; Statement $-2$ is false.

c
Let $y=x^{2} \cdot e^{-x}$

For increasing function,

$\frac{d y}{d x}>0 \Rightarrow x\left[(2-x) e^{-x}\right]>0$

$\because x>0, \therefore(2-x) e^{-x}>0$

$\Rightarrow(2-x) \frac{1}{e^{x}}>0$

For $0 < x < 2,\,\,\,(2 - x) < 0$

$\therefore \frac{1}{e^{x}}<0,$ but it is not possible

Hence the statement - $2$ is false.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from 6. Application of derivatives→6. Application of derivatives — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

The maximum value of $xy $ when $x + 2y = 8$ is

If a point $R (4, y, z)$ lies on the line segment joining the points $P (2, -3, 4)$ and $Q (8, 0, 10)$, then the distance of $R$ from the origin is

The solution of the equation $\sqrt {a + x} \frac{{dy}}{{dx}} + x = 0$ is

Let $A$ and $B$ be non empty sets in $R$ and $f : A \to B$ is a bijective function.
Statement $1$ : $f$ is an onto function .
Statement $2$ : There exists a function $g : B \to A$ such that $fog = I_B$

If $f(x) = \log \frac{{1 + x}}{{1 - x}}$, then $f(x)$ is

If G is the intersection of diagonals of a parallelogram ABCD and O is any point, then $\overrightarrow{\text{OA}}+\overrightarrow{\text{OB}}+\overrightarrow{\text{OC}}+\overrightarrow{\text{OD}}=$

$2\overrightarrow{\text{OG}}$
$4\overrightarrow{\text{OG}}$
$5\overrightarrow{\text{OG}}$
$3\overrightarrow{\text{OG}}$

If A_n be the area bounded by the curve y = (tanx)ⁿ and the lines x = 0, y = 0 and $\text{x}=\frac{\pi}{4},$ then for x > 2

$\text{A}_{\text{n}}+\text{A}_{\text{n}-2}=\frac{1}{\text{n}-1}$
$\text{A}_{\text{n}}+\text{A}_{\text{n}-2}<\frac{1}{\text{n}-1}$
$\text{A}_{\text{n}}-\text{A}_{\text{n}-2}=\frac{1}{\text{n}-1}$
none of these

$\int\frac{2\text{dx}}{\sqrt{1-4\text{x}^2}}=$

$\tan^{-1}(2\text{x})+\text{c}$
$\cot^{-1}(2\text{x})+\text{c}$
$\cos^{-1}(2\text{x})+\text{c}$
$\sin^{-1}(2\text{x})+\text{c}$

The slope of the tangent to the curve x = t² + 3 t - 8, y = 2t² - 2t - 5 at point (2, -1) is:

$\frac{22}{7}$
$\frac{6}{7}$
$-6$
$\text{None of these}$

If the matrix $\left[ {\begin{array}{*{20}{c}}1&3&{\lambda + 2}\\2&4&8\\3&5&{10}\end{array}} \right]$ is singular, then $\lambda = $