Maxima and Minima

The sum of the abosolute maximum and minimum values of the function $f\left(x\right)=\left|x^2-5x+6\right|-3x+2$ in the interval $\left[-1,3\right]$ is equal to :

The absolute minimum value, of the function $f\left(x\right)=\left|x^2-x+1\right|+\left[x^2-x+1\right]$, where $\left[t\right]$ denotes the greatest integer function, in the interval $\left[-1,2\right]$, is

Let $a_1,a_2,a_3,\dots \dots$. be an A.P. If $a_7=3$, the product $\left(a_1{a}_4\right)$ is minimum and the sum of its first $n$ terms is zero then $n!-4a_{n\left(n+2\right)}$ is equal to

A wire of length $20 \mathrm{m}$ is to be cut into two pieces. A piece of length ${\ell }_1$ is bent to make a square of area $A_1$ and the other piece of length ${\ell }_2$ is made into a circle of area $A_2$. If $2 A_1+3 A_2$ is minimum then $\left(\pi {\ell }_1\right):{\ell }_2$ is equal to:

The range of the function $f\left(x\right)=\sqrt{3-x}+\sqrt{2+x}$ is

If the functions $f\left(x\right)=\frac{x^3}{3}+2bx+\frac{a{x}^2}{2}$ and $g\left(x\right)=\frac{x^3}{3}+ax+b{x}^2,a\ne 2b$ have a common extreme point, then $a+2b+7$ is equal to

Let the function $f\left(x\right)=2x^3+(2p-7)x^2+3(2p-9)x-6$ have a maxima for some value of $\mathrm{x}<0$ and a minima for some value of $x>0$. Then, the set of all values of $p$ is

Let M be the maximum value of the product of two positive integers when their sum is $66$ . Let the sample space $S=\left\{x\in Z:x(66-x)\ge \frac{5}{9}M\right\}$ and the event $\mathrm{A}=\{\mathrm{x}\in \mathrm{S}:\mathrm{x}$ is a multiple of $3$}. Then $\mathrm{P}\left(\mathrm{A}\right)$ is equal to

Let $x=2$ be a local minima of the function $f\left(x\right)=2x^4-18x^2+8x+12,x\in \left(-4,4\right)$. If $M$ is local maximum value of the function $f$ in $\left(-4,4\right)$, then $M=$

The sum of the absolute maximum and absolute minimum values of the function $f\left(x\right)={\tan }^{-1}\left(\sin x-\cos x\right)$ in the interval $\left[0,\pi \right]$ is

The sum of the maximum and minimum values of the function $f\left(x\right)=\left|5x-7\right|+\left[x^2+2x\right]$ in the interval $\left[\frac{5}{4},2\right]$, where $\left[t\right]$ is the greatest integer $\le t$, is ______.

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

The curve $y\left(x\right)=a{x}^3+b{x}^2+cx+5$ touches the $x$-axis at the point $P\left(-2,0\right)$ and cuts the $y$-axis at the point $\mathrm{Q}$, where $y^{\text{'}}$ is equal to $3$. Then the local maximum value of $y\left(x\right)$ is

Let $f: R\to R$ be a function defined by $f\left(x\right)={\left(x-3\right)}^{n_1}{\left(x-5\right)}^{n_2}, n_1, n_2\in N$. Then, which of the following is NOT true?

The sum of the absolute minimum and the absolute maximum values of the function $f\left(x\right)=\left|3x-x^2+2\right|-x$ in the interval $\left[-1,2\right]$ is

Let $f\left(x\right)=\left|\left(x-1\right)\left(x^2-2x-3\right)\right|+x-3,x\in \mathrm{ℝ}$. If $m$ and $M$ are respectively the number of points of local minimum and local maximum of $f$ in the interval $\left(0,4\right)$, then $m+M$ is equal to _____.

Let ${\lambda }^{*}$ be the largest value of $\lambda$ for which the function $f_{\lambda }\left(x\right)=4\lambda x^3-36\lambda x^2+36x+48$ is increasing for all $x\in \mathrm{ℝ}$. Then $f_{{\lambda }^{*}}\left(1\right)+f_{\lambda ,*}\left(-1\right)$ is equal to:

Let the maximum area of the triangle that can be inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{4}=1,a>2$, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the $y$-axis, be $6\sqrt{3}$. Then the eccentricity of the ellipse is:

The sum of absolute maximum and absolute minimum values of the function $f\left(x\right)=\left|2x^2+3x-2\right|+\sin x\cos x$ in the interval $\left[0,1\right]$ is

For the function $f\left(x\right)=4{\log }_e\left(x-1\right)-2x^2+4x+5,x>1$, which one of the following is NOT correct?