Let `f(x)=-sin^3x+3sin^2x+5on[0,pi/2]` . Find the local maximum and local minimum of `f(x)dot`

Answer

Strictly increasing

Explanation

f(x) =-`sin^(3) x + 3 sin^(2) x+5`
`therefore f(X) =-3 cos x sin^(2) + 6 sinxcos x`
Now `sin x -2 lt 0 forall x in [0,(pi)/(2)]`
sinx cos `x ge 0 forall x in [0,(pi)/(2)]`
Thus f(x) is a strictly incresing function `forall x in [0,pi//2]`
Hence f(x) is minimum when x=0 and maximum when `x =pi//2`
`f_(min)=f(0)=5`
`f_(max)=f(pi//2)=7`