MCQ
The maximum value of ${x^4}{e^{ - {x^2}}}$ is
- A$e^2$
- B$e^{-2}$
- C$12e^{-2}$
- ✓$4e^{-2}$
✓
Answer
Correct option: D.
$4e^{-2}$
d
$f(x)=x^{4} e^{-x^{2}}$ or $f^{\prime}(x)=4 x^{3} e^{-x^{2}}+x^{4} e^{-x^{2}}(-2 x)$
$f(x)=x^{4} e^{-x^{2}}$ or $f^{\prime}(x)=4 x^{3} e^{-x^{2}}+x^{4} e^{-x^{2}}(-2 x)$
$2 x^{3} e^{-x^{2}}\left(2-x^{2}\right)$
Sign scheme of $f^{\prime}(x)$ is as follows:
Hence, $f(x)$ is maximum at $x=\pm \sqrt{2}.$
Thus, maximum value $=4 \mathrm{e}^{-2}.$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
In a throw of three dice, the probability that at least one die shows up $1$, is
→If $a,b,c$ are different and $\left| {\,\begin{array}{*{20}{c}}a&{{a^2}}&{{a^3} - 1}\\b&{{b^2}}&{{b^3} - 1}\\c&{{c^2}}&{{c^3} - 1}\end{array}\,} \right| = 0$, then
→A variable straight line $AB$ divides the circumference of the circle $x^2 + y^2 = 25$ in the ratio $1 : 2$ . If a tangent $CD$ is drawn to the smaller arc parallel to $AB$ , such that $ABCD$ is a rectangle, then locus of $C$ & $D$ is (as shown in the figure)
→The value of $\left( {\vec a + 2\vec b - \vec c} \right).\left\{ {\left( {\vec a - \vec b} \right) \times \left( {\vec a - \vec b - \vec c} \right)} \right\}$ is equal to
→If $u, v$ and $w$ are three non-coplanar vectors, then $(u + v - w)\,.\,[(u - v) \times (v - w)]$ equals
→Let $A$ and $B$ be any two $3\times3$ matrices. If $A$ is symmetric and $B$ is skewsymmetric, then the matrix $AB - BA$ is
→If $R(t) = \left[ {\begin{array}{*{20}{c}}{\cos t}&{\sin t}\\{ - \sin t}&{\cos t}\end{array}} \right],$then $R(s).\,R(t) = $
→The equation ${x^2} - 16xy - 11{y^2} - 12x + 6y + 21 = 0$ represents
→For $x \in R -\{0,1\},$ ${f_1}\left( x \right) = \frac{1}{x},{f_2}\left( x \right) = 1 - x$ and $f_{3}(x)=\frac{1}{1-x}$ be three given functions. If a function, $J ( x )$ satisfies $\left( {{f_2}oJo{f_1}} \right)\left( x \right)= f _{3}( x )$ then $J ( x )$ is equal to
→The number of ordered pairs $(x, y)$ of real numbers that satisfy the simultaneous equations $x+y^2=x^2+y=12$ is
→