The greatest value of f(x)=2sinx+sin2x on [0,3π/2], is given b

f'(x)=2cosx+2cos2x and f''(x)=-2sinx-4sin2xFor extreme value f'(x)=0⇒cosx+2cos2x-1=0⇒cosx=-1 or 1/2⇒x=π or π/3 as x∈[0,3π/2Now f(π)=0 and f(π/3)=232+32=332Also f(0)=0 and f(3π/2)=-2  so the greatest value of  f(x) is 332