Question

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

• A. all a > 0
• B. all b > 0
• C. all a and b
• D. all b, if a = 0

Answer 1

The correct option is D

all b, if a = 0

Since, f(x) has a relative minimum at x = 0
$\therefore$ f'(0)=0 and f'' (0) > 0
Now, y = f(x) + ax + b
$\Rightarrow \frac{dy}{dx}=f^{\prime }\left(x\right)+a\therefore {\left[\frac{dy}{dx}\right]}_{x=0}=f^{\prime }\left(0\right)+a=a=0,ifa=0Also,\frac{d^{2}y}{d{x}^2}=f^{\prime \prime }\left(x\right)\therefore {\left[\frac{d^{2}y}{d{x}^2}\right]}_{x=0}=f^{\prime \prime }\left(0\right)>0$
$\therefore$ y has a relative minimum at x = 0, if a = 0 and for all b.