Question

If $f\left(x\right)=\{\begin{array}{c}{x}^2+2,x\le 0\\ x^2+\frac{15}{8},x>0\end{array}$ then

• A. $f\left(x\right)$ increases at $x=0$
• B. $x=0$ is a point of local maximum of $f\left(x\right)$
• C. $x=0$ is a point of local minimum of $f\left(x\right)$
• D. $x=0$ is not an extremum of $f\left(x\right)$

Answer 1

Answer: (B)

$x=0$ is a point of local maximum of $f\left(x\right)$

When $x\le 0,f\left(x\right)=x^2+2$

$f^{\prime }\left(x\right)=2x⟹f^{\prime }\left(x\right)\le 0$

When $x>0,f\left(x\right)=x^2+\frac{15}{8}$

$f^{\prime }\left(x\right)=2x⟹f^{\prime }\left(x\right)>0$

$⟹x=0$ is a point of local minimum of $f\left(x\right)$