Download App

Applications of Derivative — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsApplications of Derivative1 Mark
MCQ
Function $y=x^2 e^{-x}$ is decreasing in which of interval :
  • A
    $(0,2)$
  • B
    $(2, \infty)$
  • C
    $(-\infty, 0)$
  • ✓
    $(-\infty, 0) \cup(2, \infty)$
✓

Answer

Correct option: D.
$(-\infty, 0) \cup(2, \infty)$
(D) Given $\quad y=f(x)=x^2 e^{-x}$$
\begin{aligned}
\therefore \quad f^{\prime}(x) & =x^2 e^{-x}(-1)+2 x e^{-x} \\
& =x e^{-x}(2-x)
\end{aligned}
$
now $e^{-x}>0 \quad \forall x \in R$
Hence $f(x)$ will be decreasing if$
\begin{aligned}
f^{\prime}(x)<0 \\
\Rightarrow x(2-x)<0 \\
\Rightarrow x(x-2)>0 \\
\Rightarrow &x \in(-\infty, 0) \cup(2, \infty)
\end{aligned}
$

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 Applications of Derivative→Applications of Derivative — 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 vectors, $\overrightarrow{\mathrm{p}}=(a+1) \hat{\mathrm{i}}+a \hat{\mathrm{j}}+a \hat{\mathrm{k}}$  ; $\overrightarrow{\mathrm{q}}=\mathrm{a} \hat{\mathrm{i}}+(\mathrm{a}+1) \hat{\mathrm{j}}+\mathrm{a} \hat{\mathrm{k}}$ and  $\overrightarrow{\mathrm{r}}=\mathrm{a} \hat{\mathrm{i}}+\mathrm{a} \hat{\mathrm{j}}+(\mathrm{a}+1) \hat{\mathrm{k}}(\mathrm{a} \in \mathrm{R})$ are coplanar and $3(\overrightarrow{\mathrm{p}} \cdot \overrightarrow{\mathrm{q}})^{2}-\lambda|\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{q}}|^{2}=0,$ then the value of $\lambda$ is
→
$\int_{}^{} {\frac{1}{{x\sqrt {1 + \log x} }}\;dx = } $
→
$\smallint \frac{{{{\sin }^2}x{{\cos }^2}x}}{{{{\left( {{{\sin }^5}x + {{\cos }^3}x{{\sin }^2}x + {{\sin }^3}x{{\cos }^2}x + {{\cos }^5}x} \right)}^2}}}dx$
→
Number of binary operations on the set {a, b} are:
  1. 10
  2. 16
  3. 20
  4. 8.
→
$2{\tan ^{ - 1}}\left( {\frac{1}{3}} \right) + {\tan ^{ - 1}}\left( {\frac{1}{7}} \right) = $
→
Let a random variable $X$ have a binomial distribution with mean $8$ and variance $4$. If $P\left( {X \le 2} \right) = \frac{k}{{{2^{16}}}}$,  then $k$ is equal to
→
For the multiplication of matrices as a binary operation on the set of all matrices of the form $\begin{bmatrix}\text{a}&\text{b}\\-\text{b}&\text{a}\end{bmatrix},\text{a, b}\in\text{R}$ the inverse of $\begin{bmatrix}2&3\\-3&2\end{bmatrix}$ is:
  1. $\begin{bmatrix}-2&3\\-3&-2\end{bmatrix}$
  2. $\begin{bmatrix}2&3\\-3&2\end{bmatrix}$
  3. $\begin{bmatrix}\frac{2}{13}&\frac{-3}{13}\\\frac{3}{13}&\frac{2}{13}\end{bmatrix}$
  4. $\begin{bmatrix}1&0\\0&1\end{bmatrix}$
→
For $n\, \in \,N$, the value of  $\int\limits_0^{n\pi  + V} {\sqrt {\frac{{1 + cos\ 2x}}{2}} } dx$ is  . . . (where  $\frac{\pi }{2} < \,\,V\, < \,\,\pi $ )
→
A straight line passes through (1, -2, 3) and perpendicular to the plane 2x + 3y - z = 7. Find the direction ratios of normal to plane:
  1. < 2, 3, -1 >
  2. < 2, 3, 1 >
  3. < -1, 2, 3 >
  4. None of the above
→
Suppose the function $f (x) - f (2x)$ has the derivative $5$ at $x = 1$ and derivative $7$ at $x = 2$. The derivative of the function $f (x) - f (4x)$ at $x = 1$, has the value equal to
→