Second Derivative of a Function

Q.

If a differentiable function f(x) has a relative minimum at x = 0, then the function y = f(x) + ax + b has a relative minimum at x = 0 for

1. all a > 0

2. all b > 0

3. all a and b

4. all b, if a = 0

Q. If $\frac{d}{dx}f\left(x\right)=4x^3-\frac{3}{x^4}$ such that $f\left(2\right)=0$. Then $f\left(x\right)$ is

1. $x^4+\frac{1}{x^3}-\frac{129}{8}$
2. $x^3+\frac{1}{x^4}+\frac{129}{8}$
3. $x^4+\frac{1}{x^3}+\frac{129}{8}$
4. $x^3+\frac{1}{x^4}-\frac{129}{8}$

Q. Let $f$ be a real-valued function defined on the intrval $(-1,1)$ such that $e^{-x}f\left(x\right)=2+{\int }_0^x\sqrt{t^4+1}dt$, for all $xϵ(-1,1)$, and let $f^{-1}$ be the inverse function of $f$. Then $\left(f^{-1}\right)‘\left(2\right)$ is equal to

1. $\frac{1}{2}$
2. $\frac{1}{3}$
3. $1$
4. $\frac{1}{e}$

Q.

If a differentiable function f(x) has a relative minimum at x = 0, then the function y = f(x) + ax + b has a relative minimum at x = 0 for

1. all a > 0

2. all b > 0

3. all a and b

4. all b, if a = 0

Q. Let $f$ be a function defined implicitly by the equation $\frac{1-e^{f\left(x\right)}}{1+e^{f\left(x\right)}}=x$ and $g$ be the inverse of $f.$ If $g^{\prime \prime }\left(\ln 3\right)-g^{\prime }\left(\ln 3\right)=\frac{p}{q},$ where $p$ and $q$ are relatively prime, then the value of $(p+q)$ is

Q. Let $f$ be a function defined implicitly by the equation $\frac{1-e^{f\left(x\right)}}{1+e^{f\left(x\right)}}=x$ and $g$ be the inverse of $f.$ If $g^{\prime \prime }\left(\ln 3\right)-g^{\prime }\left(\ln 3\right)=\frac{p}{q},$ where $p$ and $q$ are relatively prime, then the value of $(p+q)$ is

Q. Let $f:R\to R$ be defined as $f\left(x\right)=\{\begin{array}{ccc}{x}^5\sin \left(\frac{1}{x}\right)& +5x^2,& x<0\\ 0& & x=0\\ x^5\cos \left(\frac{1}{x}\right)& +\lambda x^2,& x>0\end{array}$ The value of $\lambda$ for which $f^{\prime \prime }\left(0\right)$ exists, is

Q. Let a function $f:(0,\infty )\to (1,\infty )$ is defined as $f\left(x\right)=x+e^{-x},$ then which of the following is correct

1. $f^{-1}\left(x\right)$ does not exist
2. $f^{-1}\left(x\right)$ is decreasing with concave upwards
3. $f^{-1}\left(x\right)$ is increasing with concave upwards
4. $f^{-1}\left(x\right)$ is increasing with concave downwards

Q. For $x\in R,x\ne 0$ if $y\left(x\right)$ is a differentiable function such that $x\underset{1}{\overset{x}{\int }}y\left(t\right)dt=(x+1)\underset{1}{\overset{x}{\int }}ty\left(t\right)dt,$ then $y\left(x\right)$ equals:

1. $C{x}^3{e}^{\frac{1}{x}}$
2. $\frac{C}{x^2}{e}^{-\frac{1}{x}}$
3. $\frac{C}{x^3}{e}^{-\frac{1}{x}}$
4. $\frac{C}{x}{e}^{-\frac{1}{x}}$

Q.

If a differentiable function f(x) has a relative minimum at x = 0, then the function y = f(x) + ax + b has a relative minimum at x = 0 for

1. all a > 0

2. all b > 0

3. all a and b

4. all b, if a = 0

Q. Let $f$ be a function defined implicitly by the equation $\frac{1-e^{f\left(x\right)}}{1+e^{f\left(x\right)}}=x.$ If $g$ be the inverse of $f,$ then which of the following options is INCORRECT ?

1. $g\left(x\right)$ is an odd function.
2. $g\left(x\right)$ is strictly decreasing function.
3. $g\left(x\right)$ is differentiable at all points.
4. Range of $g\left(x\right)$ is $[-1,1]$.