Consider `f(x)={{:(cosx,0lexlt(pi)/(2)),(((pi)/(2)-x)^(2),(pi)/(2)lexltpi):}` such that f is periodic with period `pi`. Then which of the following is not true?

Options

A. The range of f is `[0,(pi^(2))/(4))`.
B. f is discontinuous for infinite values of x.
C. The area bounded by y = f(x) and the X-axis from x = 0 to x = `npi` is `n(1+(pi^(3))/(24))` for a given `n in N`.
D. none of these

Answer

D

Explanation

Given, `={{:(cosx, 0le lt(pi)/(2)),(((pi)/(2)-x)^(2),pi//2lexltpi):}` and f is periodic with period `pi`
Graph of the function is as shown in the following figure.

From the graph, the range of the function is `[0,(pi^(2))/(2))`
It is discontinuous at `x=mpi, n in I`.
Area bounded by y = f(x) and X-axis from 0 to `npi` for `n in N`
`=n int_(0)^(pi)f(x)dx`
`=n[int_(0)^(pi//2)cosxdx+int_(pi//2)^(pi)((pi)/(2)-x)^(2)dx]=n(1+(pi^(3))/(24))`