Maxima & minima practice · Medium

The minimum value of $4{e^{2x}} + 9{e^{ - 2x}}$ is

For a certain curve $\frac{d^{2} y}{d x^{2}}=6 x-4$ and curve has local minimum value 5 at $x=1$. Let the global maximum and global minimum values, where $0 \leq x \leq 2$; are $M$ and $m$. Then the value of $(M-m)$ equals to :

The function $f\left( x \right) = a\sin x + \frac{1}{3}\sin 3x$ has maximum value at $x = \frac{\pi }{3}$. The value of $a$ is

If the maximum value of $a$, for which the function $f_a(x)=\tan ^{-1} 2 x-3 a x+7$ is non-decreasing in $\left(-\frac{\pi}{6}, \frac{\pi}{6}\right)$, is $\bar{a}$, then $f_{\bar{a}}\left(\frac{\pi}{8}\right)$ is equal to

$\max \limits_{{0 \leq x \leq \pi}}\left\{x-2 \sin x \cos x+\frac{1}{3} \sin 3 x\right\}=$

The function $f\left( x \right) = \frac{x}{2} + \frac{2}{x}$ has local minimum at

If $4 x^4+9 y^4=64$ then the maximum value of $x^2+y^2$ is (where $\mathrm{x}$ and $\mathrm{y}$ are real)

The function $y = a\left( {1 - \cos x} \right)$ is maximum when x is equal to

If $a, b>0$, then the minimum value $y=\frac{b^2}{a-x}+\frac{a^2}{x}, 0 < x < a$ is

On the interval [0, 1] the function ${x^{25}}{\left( {1 - x} \right)^{75}}$ takes its maximum value at the point

The function $f\left( x \right) = {x^4} - 62\;{x^2} + ax + 9$ attains its maximum value on the interval [0, 2] at $x = 1$. Then, the value of $a$ is

The greatest value of the function $f\left( x \right) = x{e^{ - x}}$ in $\left[ {0,\;\infty } \right]$, is

The value of a for which the function $f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {{{\tan }^{ - 1}}a - 3{x^2},\;0 < x < 1}\\ { - 6x,\;\;x \ge 1} \end{array}} \right.$ has a maximum at $x = 1$, is

Greatest and least value of $\left(\sin ^{-1} x\right)^2+\left(\cos ^{-1} x\right)^2$ are respectively

The value(s) of '$a$' for which $f(x)=a x^3+(a+6) x^2+4 x+7$ is monotonic $\forall \, x < 0$ is/are

The maximum value of $f(x)=\cos x \sqrt{\sin x}$ on the interval $[0, \pi / 2]$ is

The marks obtained by a stu first and seound terminal examinamuns are 62 antu-n 48 , respectively. The minimum marks he should get in the annual examination to have an average of at least 60 marks, are

If $f\left( x \right) = \frac{{{x^2} - 1}}{{{x^2} + 1}}$, for every real number $x$, then minimum value of $f\left( x \right)$

The denominator of a fraction is greatest then 16 of the square of numerator, then least value of fraction is

The function $f\left( x \right) = \int \limits_{ - 1}^x t\left( {{e^t} - 1} \right)\left( {t - 1} \right){\left( {t - 2} \right)^3}{\left( {t - 5} \right)^5}dt$ has local minimum at $x =$

In $\left( { - 4,\;4} \right)$ the function $f\left( x \right) = \int \nolimits_{ - 10}^x \left( {{t^4} - 4} \right){e^{ - 4t}}\;dt$ has

The real number $x$ when added to its inverse gives the minimum value of the sum at $x$ equals to

Let $f(x)=\sin x \cos ^2 x, x \in[0, \pi]$. Let $m$ be the number of values of $x$ at which $f(x)$ is minimum and $n$ be the number of values of $x$ at which $f(x)$ is maximum. Then

The minimum value of ${x^2} + \frac{1}{{1 + {x^2}}}\;$ is at

Let $f(x)=2 x+\tan ^{-1} x$ and $g(x)=\log _{e}\left(\sqrt{1+x^{2}}+x\right), x \in[0,3]$. Then

The curve $y=f(x)$ satisfies $\frac{d^{2} y}{d x^{2}}=6 x-4$ and $f(x)$ has a local minimum value 5 when $x=1$. Then $f(0)$ is equal to :

The number of critical points for the function
$f(x)=\frac{1}{3} \sin \theta \tan ^3 x+(\sin \theta-1) \tan x+\sqrt{\frac{\theta-2}{8-\theta}}$
where $\pi<\theta<2 \pi$ is

If $y=a \log |x|+b x^2+x$ has its extremum values at $x=-1$ and $x=2,$ then

Let $p\left( x \right) = {a_0} + {a_1}{x^2} + {a_2}{x^4} + \ldots + {a_n}{x^2}^n$ be a polynomial in a real variable $x$ with $0 < {a_0} < {a_1} < {a_2} \ldots < {a_n}.\;$ The function $p\left( x \right)$ has

The function $f\left( x \right) = {x^{ - x}},\;\left( {x \in R} \right)$ attains a maximum, value at $x$ which is