Applications of Derivatives

Let $f:\mathbb R\to\mathbb R$ be defined as $f(x)=|x|+|x^2-1|$. The total number of points at which $f$ attains either a local maximum or a local minimum is

Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. If $p(1)=6$ and $p(3)=2$, then $p'(0)$ is

Which of the following is true?

Consider the statements:

P: There exists some $x\in\mathbb R$ such that $f(x)+2x=2(1+x^2)$

Q: There exists some $x\in\mathbb R$ such that $2f(x)+1=2x(1+x)$

Then

If $f(x)=\displaystyle\int_0^x e^{t^2}(t-2)(t-3)\,dt$ for all $x\in(0,\infty)$, then

The number of distinct real roots of $x^4-4x^3+12x^2+x-1=0$ is

Match the statements given in Column I with the intervals/union of intervals given in Column II

Let f, g and h be real-valued functions defined on the interval [0, 1] by $f(x)=e^{x^2}+e^{-x^2}$, $g(x)=xe^{x^2}+e^{-x^2}$ and $h(x)=x^2e^{x^2}+e^{-x^2}$. If a, b and c denote, respectively, the absolute maximum of f, g and h on [0, 1], then

Let f be a real-valued differentiable function on $\mathbb R$ (the set of all real numbers) such that $f(1)=1$. If the y-intercept of the tangent at any point P(x, y) on the curve $y=f(x)$ is equal to the cube of the abscissa of P, then the value of $f(-3)$ is equal to

Let $f$ be a function defined on $\mathbb R$ (the set of all real numbers) such that $f'(x)=2010(x-2009)(x-2010)^2(x-2011)^3(x-2012)^4$, for all $x\in\mathbb R$.

If $g$ is a function defined on $\mathbb R$ with values in the interval $(0,\infty)$ such that $$f(x)=\ln(g(x)),\ \text{for all } x\in\mathbb R,$$ then the number of points in $\mathbb R$ at which $g$ has a local maximum is

The real number $s$ lies in the interval

The function $f'(x)$ is

Match the statements/expressions in Column I with the open intervals in Column II.

For the function
$$f(x)=x\cos\frac1x,\qquad x\ge1,$$

Match the statements/expressions given in Column I with the values given in Column II.

The maximum value of the function $f(x)=2x^3-15x^2+36x-48$ on the set $A=\left\{x\mid x^2+20\le9x\right\}$ is

Let $p(x)$ be a polynomial of degree 4 having extremum at $x=1,2$ and
$$\lim_{x\to0}\left(1+\frac{p(x)}{x^2}\right)=2.$$
Then the value of $p(2)$ is

The total number of local maxima and local minima of the function
$$f(x)=\begin{cases}(2+x)^3,&-3<x\le-1\\x^{2/3},&-1<x<2\end{cases}$$
is

Let $f(x)$ be a non-constant twice differentiable function defined on $(-\infty,\infty)$ such that $f(x)=f(1-x)$ and $f'\left(\frac14\right)=0$. Then,

Let the function $g:(-\infty,\infty)\to\left(-\frac\pi2,\frac\pi2\right)$ be given by $g(u)=2\tan^{-1}(e^u)-\frac\pi2$. Then, $g$ is