Let $(2,3)$ be the largest open interval in which the function $f(x)=2{\log }_{\mathrm{e}}(x-2)-x^2+ax+1$ is strictly increasing and (b, c) be the largest open interval, in which the function $\mathrm{g}(x)=(x-1{)}^3(x+2-\mathrm{a}{)}^2$ is strictly decreasing. Then $100(\mathrm{a}+\mathrm{b}-\mathrm{c})$ is equal to :

The sum of all local minimum values of the function $$\mathrm{f}(x)=\{\begin{array}{lr}1-2x,& x2\end{array}$$ is

Let the function $f(x)=\frac{x}{3}+\frac{3}{x}+3,x\ne 0$ be strictly increasing in $(-\infty ,{\alpha }_1)\cup ({\alpha }_2,\infty )$ and strictly decreasing in $({\alpha }_3,{\alpha }_4)\cup ({\alpha }_4,{\alpha }_5)$. Then $\sum _{i=1}^5{\alpha }_i^2$ is equal to

If the function $f(x)=2x^3-9a{x}^2+12{\mathrm{a}}^{2}x+1$, where $\mathrm{a}>0$, attains its local maximum and local minimum values at p and q , respectively, such that ${\mathrm{p}}^2=\mathrm{q}$, then $f(3)$ is equal to :

Let f : ℝ $$\to$$ ℝ be a polynomial function of degree four having extreme values at x = 4 and x = 5. If $\underset{x\to 0}{\lim }\frac{f(x)}{x^2}=5$, then f(2) is equal to :

Let $x=-1$ and $x=2$ be the critical points of the function $f(x)=x^3+a{x}^2+b{\log }_{\mathrm{e}}|x|+1,x\ne 0$. Let $m$ and M respectively be the absolute minimum and the absolute maximum values of $f$ in the interval $\left[-2,-\frac{1}{2}\right]$. Then $|\mathrm{M}+m|$ is equal to $($ Take ${\log }_{\mathrm{e}}2=0.7):$

Let $\mathrm{a}>0$. If the function $f(x)=6x^3-45\mathrm{a}{x}^2+108{\mathrm{a}}^{2}x+1$ attains its local maximum and minimum values at the points $x_1$ and $x_2$ respectively such that $x_1{x}_2=54$, then $\mathrm{a}+x_1+x_2$ is equal to :

The shortest distance between the curves $y^2=8x$ and $x^2+y^2+12y+35=0$ is:

Let $f:\mathrm{R}\to \mathrm{R}$ be a function defined by $f(x)=||x+2|-2| x \|$. If $m$ is the number of points of local minima and $n$ is the number of points of local maxima of $f$, then $m+n$ is

Let the area enclosed between the curves $|y|=1-x^2$ and $x^2+y^2=1$ be $\alpha$. If $9\alpha =\beta \pi +\gamma ;\beta ,\gamma$ are integers, then the value of $|\beta -\gamma |$ equals:

Let the area of the region $(x,y):2y\le x^2+3,y+|x|\le 3,y\ge |x-1|$ be $ A $. Then $ 6A $ is equal to :

The area of the region, inside the circle $(x-2\sqrt{3}{)}^2+y^2=12$ and outside the parabola $y^2=2\sqrt{3}x$ is :

The area of the region enclosed by the curves $y=x^2-4x+4$ and $y^2=16-8x$ is :

The area of the region bounded by the curves $x(1+y^2)=1$ and $y^2=2x$ is:

If the area of the region $\left\{(x,y):-1\le x\le 1,0\le y\le \mathrm{a}+{\mathrm{e}}^{|x|}-{\mathrm{e}}^{-x},\mathrm{a}>0\right\}$ is $\frac{{\mathrm{e}}^2+8\mathrm{e}+1}{\mathrm{e}}$, then the value of $a$ is :

The area of the region $\left\{(x,y):x^2+4x+2\le y\le |x+2|\right\}$ is equal to

The area of the region enclosed by the curves $y={\mathrm{e}}^x,y=\left|{\mathrm{e}}^x-1\right|$ and $y$-axis is :

The area (in sq. units) of the region $\left\{(x,\mathrm{y}):0\le \mathrm{y}\le 2|x|+1,0\le \mathrm{y}\le x^2+1,|x|\le 3\right\}$ is

If the area of the region $\{(x,y):1+x^2\le y\le \min \{x+7,11-3x\}\}$ is $ A $, then $ 3A $ is equal to :

If the area of the region bounded by the curves $y=4-\frac{x^2}{4}$ and $y=\frac{x-4}{2}$ is equal to $\alpha$, then $6\alpha$. equals