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 the lines $\frac{x-k}{1}=\frac{y-2}{2}=\frac{z-3}{3}$ and $\frac{x+1}{3}=\frac{y+2}{2}=\frac{z+3}{1}$ are co-planar, then the value of $k$ is $.....$

$\sin \left[ {\frac{\pi }{2} - {{\sin }^{ - 1}}\left( { - \frac{{\sqrt 3 }}{2}} \right)} \right] = $

Let $A =$ $\left[ {\begin{array}{*{20}{c}}{x + \lambda }&x&x\\x&{x + \lambda}&x\\x&x&{x + \lambda }\end{array}} \right]$ , then $A^{-1}$ exists if

For $x \in R$, let the function $y(x)$ be the solution of the differential equation

$\frac{d y}{d x}+12 y=\cos \left(\frac{\pi}{12} x\right), y(0)=0 .$

Then, which of the following statements is/are $TRUE$?

The equation of the curve aatisfying the differential $\text{y}(\text{x}+\text{y}^{3})\text{dx}=\text{x}(\text{y}^{3}-\text{x})$ dy and passing through the point (1, 1) is:

$\text{y}^{3}-2\text{x}+3\text{x}^{2}\text{y}=0$
$\text{y}^{3}+2\text{x}+3\text{x}^{2}\text{y}=0$
$\text{y}^{3}+2\text{x}-3\text{x}^{2}\text{y}=0$
None of these.

Inverse of the matrix $\left[ {\begin{array}{*{20}{c}}3&{ - 2}&{ - 1}\\{ - 4}&1&{ - 1}\\2&0&1\end{array}} \right]$ is

Let $f$ and $g$ be differentiable functions on $\mathrm{R}$ such that $fog$ is the identity function. If for some $a, b \in \mathrm{R}, g^{\prime}(a)=5$ and $g(a)=b,$ then $f^{\prime}(b)$ is equal to

The general solution of the differential equation $y d x-x d y=0$; (Given $x, y>0)$, is of the form

The unit vector parallel to the resultant vector of $2i + 4j - 5k$ and $i + 2j + 3k$ is

$\int\tan^{-1}\sqrt{\text{xdx}}$ is equal to:

$(\text{x}+1)\tan^{-1}\sqrt{\text{x}}-\sqrt{\text{x}}+\text{c}$
$\text{x}\tan^{-1}\sqrt{\text{x}}-\sqrt{\text{x}}+\text{c}$
$\sqrt{\text{x}}-\text{x}\tan^{-1}\sqrt{\text{x}}+\text{c}$
$\sqrt{\text{x}}-(\text{x}+1)\tan^{-1}\sqrt{\text{x}}+\text{c}$