Q. If a differentiable function $f(x)$ has a relative minimum at $x = 0$, then the function $\phi(x) = f(x) + ax + b$ has a relative minimum at $x = 0$ for
A) All $a$ and all $b$
B) All $b$ if $a = 0$
C) All $b > 0$
D) All $a > 0$
Explanation
(b)
$\phi^{'}(x) = f^{'}(x) + a$
$\because\ \ \phi^{'}(0) = 0$
$\Rightarrow \ \ \ f^{'}(0) + a = 0$
$\Rightarrow \ \ a = 0\ \ \ \ (\because f^{'}(0) = 0)$
Also, $\phi^{'}(0) > 0\ \ \ \ (\because f^{''}(0) > 0)$
$\Rightarrow \ \phi^{'}(x)$ has relative minimum at $x = 0$ for all $b$ if $a = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.