Maxima & minima practice · Mixed difficulty

Let a function $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as : $f(x)=\left\{\begin{array}{cc}\int_0^x(5-|t-3|) d t, & x>4 \\ x^2+b x, & x \leq 4\end{array}\right.$
where $b \in \mathbb{R}$. If $f$ is continuous at $x=4$ then which of the following statements is NOT true?

If the cubic polynomial $y=a x^{3}+b x^{2}+c x+d(a, b, c, d \in R)$ has only one critical point in its entire domain and $a c=2$, then the value of $|b|$ is :

A covered box of volume $72 \mathrm{~cm}^3$ and the base sides in a ratio of $1: 2$ is to be made. The length all sides so that the total surface area is the least possible is

The largest value of $2{x^3} - 3{x^2} - 12x + 5\;$ for $- 2 \le x \le 4$ occurs at $x$ is equal to

Local maximum value of the function $\frac{{\log \,x}}{x}$ is

If $y = \frac{{\sin \left( {x + a} \right)}}{{\sin \left( {x + b} \right)}},a \ne b,$ then $y$ is

The function $f\left( x \right) = {x^2}{e^{ - 2x}},\;x > 0$ Then, the maximum value of $f\left( x \right)$ is

The corner points of the feasible region determined by the following system of linear inequalities:
$2 x+y \leq 10, x+3 y \leq 15, x, y \geq 0$ are $(0,0),(5,0),(3,4)$ and $(0,5)$. Let $Z=p x+q y$, where $p, q > 0$. Condition on $p$ and $q$ so that the maximum of $Z$ occurs at both $(3,4)$ and $(0,5)$ is

If $f(x)=\left\{\begin{array}{ll}2 x^2+\frac{2}{x^2}, & 0<|x| \leq 2 \\ 3, & x>2\end{array}\right.$, then

Let $\mathrm{k}$ and $\mathrm{K}$ be the minimum and the maximum values of the function $\mathrm{f}(x)=\frac{(1+x)^{0.6}}{1+x^{0.6}}$ in $[0,1]$ respectively, then the ordered pair $(\mathrm{k}, \mathrm{K})$ is equal to :

The number of values of $x$ where $f\left( x \right) = \cos x + \cos \sqrt 2 x\;$ attains its maximum is

If $\sin \theta+\cos \theta=1$, then the minimum value of $(1+\operatorname{cosec} \theta)(1+\sec \theta)$ is

The set of all values of the parameter $a$ for which the points of minimum of the function $y = 1 + {a^2}x - {x^3}$ satisfy the inequality $\frac{{{x^2} + x + 2}}{{{x^2} + 5x + 6}} \le 0$, is

If $xy = {a^2}\;\mathrm{and}\;S = {b^2}x + {c^2}y$ where $a,b\,and\,c$ are constants, then the minimum value of $S$ is

If $y = y\left( x \right)$ is the solution of the differential equation $x\frac{{dy}}{{dx}} + 2y = x{e^x},\,\,y\left( 1 \right) = 0$ then the local maximum value of the function $z\left( x \right) = {x^2}y\left( x \right) - {e^x},\,x \in R$ is

The minimum value of $px + qy$ when $xy = {r^2}$, is

If $f(x)=K x^3-9 x^2+9 x+3$ is increasing on $R$, then -

The maximum value of $f\left( x \right) = 3{\cos ^2}x + 4{\sin ^2}x + \cos \frac{x}{2} + \sin \frac{x}{2}$, is

If $x + y = 8$, then maximum value of ${x^2}y$ is

The function $f\left( x \right) = x + \frac{1}{x}$ has

If $a, b \in R$ distinct numbers satisfying $|a-1|+|b-1|=|a|+|b|=|a+1|+|b+1|$, then the minimum value of $|a-b|$ is

Let $M$ and $m$ be respectively the absolute maximum and the absolute minimum values of the function, $f(x)=2 x^3-9 x^2+12 x+5$ in the interval $[0,3]$. Then $M-m$ is equal to

A cubic $f\left( x \right)$ vanishes at $x = - 2$ and has relative minimum/maximum at $x = - 1\,and\,x = \frac{1}{3}$ such that $\int \limits_{ - 1}^1 f\left( x \right)dx = \frac{{14}}{3}$ Then, $f\left( x \right)$ is

Let $f(x)$ be a polynomial of degree 5 such that $x= \pm 1$ are its critical points. If $=4$, then which one of the following is not true ?

If ${a^2}{x^4} + {b^2}{y^4} = {c^6}$, then maximum value of $xy$ is&nbsp;

A wire of length $20 \mathrm{m}$ to be cut into two pieces. A piece of length $l_{1}$ is bent to make a square of area $A_{1}$ and the other piece of length $l_{2}$ is made into a circle of area $A_{2}$. If $2 A_{1}+3 A_{2}$ is minimum then $\left(\pi l_{1}\right): l_{2}$ is equal to

If $m$ and $M$ respectively denote the minimum and maximum of $f\left( x \right) = {\left( {x - 1} \right)^2} + 3\;for\;x \in \left[ { - 3,1} \right],$ then the ordered pair ($m,M$) is equal to

For $x \in(0, \pi)$, the least value of $f(x)=\left(\frac{2+\cos x}{\sin x}\right)^2$ is

The maximum value of $xy$ when $x + 2y = 8$ is

If $m$ and $n$ are positive integers and $f(x)=\int_1^x(t-a)^{2 n}(t-b)^{2 m+1} d t, a \neq b$, then