f(x)=0.5*e^x(sinx+cosx)
對x求導(dǎo),得
f'(x)=0.5*[e^x(sinx+cosx)+e^x(cosx-sinx)]
=0.5*e^x(2cosx)
在區(qū)間[0,π/2]上,cosx＞0,e^x＞0
∴f'(x)＞0
∴在此區(qū)間上函數(shù)遞增,
∴x=0時,函數(shù)取得最小值,最小值是f(0)=0.5
x=π/2時,函數(shù)取得最大值,最大值是f(π/2)=0.5*e^(π/2)*1=0.5*e^(π/2)
即函數(shù)f(x)在區(qū)間[0,π/2]上的值域是[0.5,0.5*e^(π/2)]