Question

Let $f\left(x\right)=a{x}^2-b|x|$, where $a$ and $b$ are constants. Then at $x=0,f\left(x\right)$ has

• A. A maxima whenever $a>0,b>0$
• B. A maxima whenever $a>0,b<0$
• C. A minima whenever $a>0,b<0$
• D. Neither minima nor a maxima whenever $a>0,b<0$

Answer 1

Answer: (A), (C)

A maxima whenever $a>0,b>0$
A minima whenever $a>0,b<0$

Given : $f\left(x\right)=a{x}^2-b\left|x\right|$

To find at $x=0f\left(x\right)=?$

$f\left(x\right)=a{x}^2-b\left|x\right|$

Let us solve this with the help of Graph

i) $a>0,b>0$

Let us first compute graph of $f\left(x\right)=a{x}^2-bx$

At $a>0,b>0$ The graph of $f\left(x\right)=a{x}^2-b\left|x\right|$

According to the graph at $x=0$ is the point of maxima.

Therefore option A is correct answer.

ii) $a>0,b=0$

Since according to the graph there aren't any maxima.

$\because x=0$ is a point of minima.

Therefore option C is also correct.

Therefore option A and C are correct.