Question

$\text{ Consider }f(x)=\{\begin{array}{ll}\cos x& 0\le x<\frac{\pi }{2}\\ {\left(\frac{\pi }{2}-x\right)}^2& \frac{\pi }{2}\le x<\pi \end{array}$ such that $f$ is periodic with period $\pi$, then

• A. $\text{ The range of }f\text{ is }\left[0,\frac{{\pi }^2}{4}\right)$
• B. fis continuous for all real $x$, but not differentiable for some real $x$
• C. fis continuous for all real $x$
• D. The area bounded by $y=f(x)$ and the $X$-axis from $x=-n\pi$ to $x=n\pi$ is $2n\left(1+\frac{{\pi }^3}{24}\right)$ for a given $n\in N$

Answer 1

Answer: (A), (D)

Given $f(x)=\{\begin{array}{ll}\cos x& 0\le x<\frac{\pi }{2}\\ {\left(\frac{\pi }{2}-x\right)}^2& \pi /2\le x<\pi \end{array}$ and $f$ is periodic with period $\pi$
Let us draw the graph of $y=f(x)$
From the graph, the range of the function is $\left[0,\frac{{\pi }^2}{4}\right)\Rightarrow$(A)
It is discontinuous at $x=n\pi ,n\in I$. It is not differentiable at $x=\frac{n\pi }{2},n\in I$.