The function f defined by f(x) = (x + 2) e^ - x is - Vidyadip

MCQ

The function $f$ defined by $f(x) = (x + 2){e^{ - x}}$ is

A
Decreasing for all $x$
B
Decreasing in $( - \infty ,\, - 1)$ and increasing in $( - 1,\infty )$
C
Increasing for all $x$
✓
Decreasing in $( - 1,\,\infty )$ and increasing in $( - \infty ,\, - 1)$

✓

Answer

Correct option: D.

Decreasing in $( - 1,\,\infty )$ and increasing in $( - \infty ,\, - 1)$

d
(d) $f(x) = (x + 2){e^{ - x}}$

$f'(x) = {e^{ - x}} - {e^{ - x}}(x + 2)$

$f'(x) = - {e^{ - x}}[x + 1]$

For increasing, $ - {e^{ - x}}(x + 1) > 0$ or ${e^{ - x}}(x + 1) < 0$

${e^{ - x}} > 0$ $(x + 1) < 0$

$x \in ( - \infty ,\,\infty )$ and $x \in ( - \infty , - 1)$

$\therefore x \in ( - \infty , - 1)$

Hence, the function is increasing in $( - \infty ,\, - 1)$

For decreasing, $ - {e^{ - x}}(x + 1) < 0$ or ${e^{ - x}}(x + 1) > 0$, $x \in ( - 1,\,\infty )$

Hence the function is decreasing in $( - 1,\;\infty )$.

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

If $A$ is a square matrix of order 3 and $|A|=5$, then the value of $\left|2 A^{\prime}\right|$ is

$\int_0^a {f(x)\,dx} = $

A zero vector has:

Any direction
No direction
Many direction
None of these

Let $f:(0, \infty) \rightarrow R$ be given by $f(x)=\int_{\frac{1}{x}}^x e^{-\left(t+\frac{1}{t}\right)} \frac{d t}{t}$. Then

$(A)$ $f(x)$ is monotonically increasing on $[1, \infty)$

$(B)$ $f(x)$ is monotonically decreasing on $(0,1)$

$(C)$ $f(x)+f\left(\frac{1}{x}\right)=0$, for all $x \in(0, \infty)$

$(D)$ $f\left(2^x\right)$ is an odd function of $x$ on $R$

Suppose that two cards are drawn at random from a deck of cards. Let $X$ be the number of aces obtained. Then the value of $\mathrm{E}(\mathrm{X})$ is

A function $f$ is defined on $[-3,3]$ as

$f(x)=\left\{\begin{array}{cc}\min \left\{|x|, 2-x^{2}\right\} & , \quad-2 \leq x \leq 2 \\ {[|x|]} & , \quad 2<|x| \leq 3\end{array}\right.$

where $[x]$ denotes the greatest integer $\leq x .$ The number of points, where $f$ is not differentiable in $(-3,3)$ is

$\int\sqrt{\frac{\text{x}}{1-\text{x}}}\text{ dx}$ is equal to:

$\sin^{-1}\sqrt{\text{x}}+\text{C}$
$\sin^{-1}\Big\{\sqrt{\text{x}}-\sqrt{\text{x}(1-\text{x})}\Big\}+\text{C}$
$\sin^{-1}\Big\{\sqrt{\text{x}(1-\text{x})}\Big\}+\text{C}$
$\sin^{-1}\sqrt{\text{x}}-\sqrt{\text{x}(1-\text{x})}+\text{C}$

If $f (a + b - x)$ $= f (x)$ , then $\int\limits_a^b {\,x.f\,(a + b - x)\,dx} $ $=$

$f(x) = \left| {\begin{array}{*{20}{c}}
{{x^3}}&{{x^2}}&{3{x^2}}\\
1&{ - 6}&4\\
p&{{p^2}}&{{p^3}}
\end{array}} \right|$ , here $ p $ is a constant, then ${{{d^3}f(x)} \over {d{x^3}}}$ is

The taxi fare in a city is as follows. For the first km the fare is Rs.10 and subsequent distance is Rs.6/ km.Taking the distance covered as x km and fare as Rs y, write a linear equation.

y = 4 + 6x
y = 4 + 5x
y = 3 + 6x
y = 3 + 5x