MCQ
In $( - 4,\,4)$ the function $f(x) = \int\limits_{ - 10}^x {({t^4} - 4){e^{ - 4t}}dt} $ has
- ANo extrema
- BOne extremum
- ✓Two extrema
- DFour extrema
✓
Answer
Correct option: C.
Two extrema
c
(c) $f(x) = \int_{ - 10}^x {({t^4} - 4){e^{ - 4t}}dt} $ ==> $f'(x) = ({x^4} - 4){e^{ - 4x}}$
(c) $f(x) = \int_{ - 10}^x {({t^4} - 4){e^{ - 4t}}dt} $ ==> $f'(x) = ({x^4} - 4){e^{ - 4x}}$
Now $f'(x) = 0 \Rightarrow x = \pm \sqrt 2 ,\, \pm \sqrt 2 $
Now $f''(x) = - \,4({x^4} - 4){e^{ - 4x}} + 4{x^3}{e^{ - 4x}}$
At $x = \sqrt 2 $ and $x = - \sqrt 2 $ the given function has extreme value.
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
If $y = {x^2} + {x^{\log x}},$ then ${{dy} \over {dx}} = $
→If $A=\left[\begin{array}{rr}2 & 1 \\ -1 & 2\end{array}\right], B=\left[\begin{array}{rr}1 & -2 \\ 2 & 1\end{array}\right], C=\left[\begin{array}{rr}1 & -3 \\ 2 & 1\end{array}\right]$,then
→The length of perpendicular from the origin to the plane which makes intercepts $\frac{1}{3},\frac{1}{4},\frac{1}{5}$ respectively on the coordinate axes is:
→A bag contains six red four green and eight white balls If a ball is picked at random the probability that it is not white is:
$(i)$ $f (x)$ is continuous and defined for all real numbers
→$(ii)$ $f '(-5) = 0 \,; \,f '(2)$ is not defined and $f '(4) = 0$
$(iii)$ $(-5, 12)$ is a point which lies on the graph of $f (x)$
$(iv)$ $f ''(2)$ is undefined, but $f ''(x)$ is negative everywhere else.
$(v)$ the signs of $f '(x)$ is given below
On the possible graph of $y = f (x)$ we have 
If $\int_{}^{} {(\cos x - \sin x)\;dx = \sqrt 2 \sin (x + \alpha ) + c} $, then $\alpha = $
→Evaluate: $\int \sec ^2(7-4 x) d x$
→If ${a^2} + {b^2} + {c^2} + ab + bc + ca \leq 0\,\forall a,\,b,\,c\, \in \,R$ , then the value of determinant $\left| {\begin{array}{*{20}{c}}
{{{(a + b + c)}^2}}&{{a^2} + {b^2}}&1 \\
1&{{{(b + c + 2)}^2}}&{{b^2} + {c^2}} \\
{{c^2} + {a^2}}&1&{{{(c + a + 2)}^2}}
\end{array}} \right|$
→{{{(a + b + c)}^2}}&{{a^2} + {b^2}}&1 \\
1&{{{(b + c + 2)}^2}}&{{b^2} + {c^2}} \\
{{c^2} + {a^2}}&1&{{{(c + a + 2)}^2}}
\end{array}} \right|$
The differential coefficient of ${\tan ^{ - 1}}\left( {{{\sqrt {1 + {x^2}} - 1} \over x}} \right)$ with respect to ${\tan ^{ - 1}} x$ is
→If $f(x) = \left\{ \begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,1,\,{\rm{when\,\,}}\,\,0 < x \le \frac{{3\pi }}{4}\\2\sin \frac{2}{9}x,{\rm{when\,\,}}\,\frac{{3\pi }}{4} < x < \pi \end{array} \right.$, then
→