If $f(x)=\left\{\begin{aligned} \sin \left(\frac{\pi x}{2}\right), & \text { if } x \lt 1 \\ 3-2 x, & \text { if } x \geq 1\end{aligned}\right.$, then $f(x)$ has

• A

  local minimum at $x=1$
• Correct answer
  B

  local maximum at $x=1$
• C

  Both local maximum and local minimum at $x=1$
• D

  None of the above

Solution

Given function is $f(x)=\left\{\begin{array}{cl}\sin \left(\frac{\pi x}{2}\right), & \text { if } x \lt 1 \\ 3-2 x, & \text { if } x \leq 1\end{array}\right.$ $\because$ LHL and RHL of $f(x)$ is 1 , when $x \rightarrow 1$ and $f(1)=3-2=1$ $\therefore f(x)$ is continuous at $x=1$. Now, graph of function $f(x)$ is
From the above graph of function, we see that $f(x)$ has local maxima at $x=1$.

Hence, the correct answer is option: 'B'.