Question

Let $f\left(x\right)=\left\{\begin{array}{l}\sin \left(\frac{\mathrm{\pi x}}{2}\right),0\le x\le 1\\ 3-2x,x\ge 1\end{array}\right.$then

• A. $f\left(x\right)$ has a local maxima at $x=1$
• B. $f\left(x\right)$ has a local minima at $x=1$
• C. $f\left(x\right)$ does not have any local extrema at $x=1$
• D. None of these

Answer 1

The correct option is A

$f\left(x\right)$ has a local maxima at $x=1$

Explanation for the correct option:

Find the characteristics of $f\left(x\right)$:

Given,

$f\left(x\right)=\left\{\begin{array}{l}\sin \left(\frac{\mathrm{\pi x}}{2}\right),0\le x\le 1\\ 3-2x,x\ge 1\end{array}\right.$

If $x=0$, $\Rightarrow f\left(x\right)=\sin \left(\frac{\pi \left(0\right)}{2}\right)=0$

If $x=1$,$\Rightarrow f\left(x\right)=\sin \left(\frac{\pi \left(1\right)}{2}\right)=1$

If $x=2$,$\Rightarrow f\left(x\right)=3-2\left(2\right)=-1$

If $x=3$, $\Rightarrow f\left(x\right)=3-2\left(3\right)=-3$

Therefore, $f\left(x\right)$ is a decreasing function and at $x=1$, we get a local maximum.

Therefore, the correct answer is option (A).