Maxima & minima practice · Mixed difficulty

Let $f(x)=x^4+a x^3+3 x^2+b x+1, a, b \in R$. If $f(x) \geq 0, \forall x \in R$, then the maximum value of $a^2+b^2$ is equal to

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

The maximum value of $f\left( x \right) = \frac{x}{{4 + x + {x^2}}}$ on [-1,1] 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

If $\alpha=2 \beta=3 \gamma$, then find the value of $\log _{\left(\frac{7}{6}\right)}\left(\frac{\alpha^2+\beta^2+\gamma^2}{\alpha \beta+\beta \gamma+\gamma \alpha}\right)$.

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

For any real $\theta$, the maximum value of $\cos ^2(\cos \theta)+\sin ^2(\sin \theta)$ is

Divide 12 into two parts such that the product of the square on one part and the fourth power of the second port is maximum, are

If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $f\left( x \right) = \int\limits_0^{{x^2}} {\frac{{{t^2} - 5t + 4}}{{2 + {e^t}}}} \,dt,$ then the ordered pair $\left( {m,n} \right)$ is equal to

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 maximum value ${x^3} - 3x$ in the interval [0, 2] is

The maximum value of ${\left( {\frac{1}{x}} \right)^{2{x^2}}}$, is

On the curve $x^3=12 y$, the values of $x$ for which the abscissa changes at a faster rate than the ordinate is

The minimum value of $2x + 3y$, when $xy = 6$, 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

For the function $f\left( x \right) = x{e^x}$ the point

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

Let $f(x)=\left\{\begin{array}{cc}x^3-x^2+10 x-7, & x \leq 1 \\ -2 x+\log _2\left(b^2-4\right), & x>1\end{array}\right.$ Then the set of all values of $b$, for which $f(x)$ has maximum value at $x=1$, is

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