A software company sets up m number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $$\mathrm{m}$$ is equal to:

If $5f(x)+4f\left(\frac{1}{x}\right)=x^2-2,\forall x\ne 0$ and $y=9x^{2}f(x)$, then $y$ is strictly increasing in :

Let $$g(x)=3f\left(\frac{x}{3}\right)+f(3-x)$$ and $$f^{\prime \prime }(x)>0$$ for all $$x\in (0,3)$$. If $$g$$ is decreasing in $$(0,\alpha )$$ and increasing in $$(\alpha ,3)$$, then $$8\alpha$$ is :

Let $$f:\to \mathbb{R}\to (0,\infty )$$ be strictly increasing function such that \lim _\limits{x \rightarrow \infty} \frac{f(7 x)}{f(x)}=1. Then, the value of \lim _\limits{x \rightarrow \infty}\left[\frac{f(5 x)}{f(x)}-1\right] is equal to

If the function $$f:(-\infty ,-1]\to (a,b]$$ defined by $$f(x)=e^{x^3-3x+1}$$ is one - one and onto, then the distance of the point $$P(2b+4,a+2)$$ from the line $$x+e^{-3}y=4$$ is :

Consider the function $$f:\left[\frac{1}{2},1\right]\to \mathbb{R}$$ defined by $$f(x)=4\sqrt{2}{x}^3-3\sqrt{2}x-1$$. Consider the statements (I) The curve $$y=f(x)$$ intersects the $$x$$-axis exactly at one point. (II) The curve $$y=f(x)$$ intersects the $$x$$-axis at $$x=\cos \frac{\pi }{12}$$. Then

The function $$f(x)=\frac{x}{x^2-6x-16},x\in \mathbb{R}-\{-2,8\}$$

The function $$f(x)=2x+3(x{)}^{\frac{2}{3}},x\in \mathbb{R}$$, has

Let $$f(x)=(x+3{)}^2(x-2{)}^3,x\in [-4,4]$$. If $$M$$ and $$m$$ are the maximum and minimum values of $$f$$, respectively in $$[-4,4]$$, then the value of $$M-m$$ is

The maximum area of a triangle whose one vertex is at $$(0,0)$$ and the other two vertices lie on the curve $$y=-2x^2+54$$ at points $$(x,y)$$ and $$(-x,y)$$, where $$y>0$$, is :

Let the sum of the maximum and the minimum values of the function $$f(x)=\frac{2x^2-3x+8}{2x^2+3x+8}$$ be $$\frac{m}{n}$$, where $$\gcd (\mathrm{m},\mathrm{n})=1$$. Then $$\mathrm{m}+\mathrm{n}$$ is equal to :

Let $$f(x)=3\sqrt{x-2}+\sqrt{4-x}$$ be a real valued function. If $$\alpha$$ and $$\beta$$ are respectively the minimum and the maximum values of $$f$$, then $${\alpha }^2+2{\beta }^2$$ is equal to

If the function $$f(x)=2x^3-9{\mathrm{ax}}^2+12{\mathrm{a}}^{2}x+1,\mathrm{a}>0$$ has a local maximum at $$x=\alpha$$ and a local minimum at $$x={\alpha }^2$$, then $$\alpha$$ and $${\alpha }^2$$ are the roots of the equation :

Let $$f(x)=4{\cos }^{3}x+3\sqrt{3}{\cos }^{2}x-10$$. The number of points of local maxima of $$f$$ in interval $$(0,2\pi )$$ is

The number of critical points of the function $$f(x)=(x-2{)}^{2/3}(2x+1)$$ is

For the function $$f(x)=(\cos x)-x+1,x\in \mathbb{R}$$, between the following two statements (S1) $$f(x)=0$$ for only one value of $$x$$ in $$[0,\pi ]$$. (S2) $$f(x)$$ is decreasing in $$\left[0,\frac{\pi }{2}\right]$$ and increasing in $$\left[\frac{\pi }{2},\pi \right]$$.

Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices of the rectangle ABCD lie on the sides of the rectangle PQRS. Let a and b be the sides of the rectangle PQRS when its area is maximum. Then (a+b)$${}^2$$ is equal to :

Let $$f(x)=x^5+2x^3+3x+1,x\in \mathbf{R}$$, and $$g(x)$$ be a function such that $$g(f(x))=x$$ for all $$x\in \mathbf{R}$$. Then $$\frac{g(7)}{g^{\prime }(7)}$$ is equal to :

For the function $$f(x)=\sin x+3x-\frac{2}{\pi }\left(x^2+x\right),\text{ where }x\in \left[0,\frac{\pi }{2}\right],$$ consider the following two statements : (I) $$f$$ is increasing in $$\left(0,\frac{\pi }{2}\right)$$. (II) $$f^{\prime }$$ is decreasing in $$\left(0,\frac{\pi }{2}\right)$$. Between the above two statements,