Second Derivative of a Function

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 $f$ 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.

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 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