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

Answer 1

Answer: (B)

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