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