由题一有f(x)=sin(x+π/4)求函数g(x)=f(x+π/4)*f(x-π/4)在区间[0,π/2]上的最大值及相应的x的值

Question

由题一有 f(x)=sin(x+π/4)
求函数g(x)=f(x+π/4)*f(x-π/4)在区间[0,π/2]上的最大值及相应的x的值

Answer 1

f(x)=sin(x+π/4)
f(x+π/4)=sin(x+π/4+π/4)=sin(x+π/2)=cosx
f(x-π/4)=sin(x-π/4+π/4)=sinx
所以g(x)=f(x+π/4)*f(x-π/4)=sinxcosx=(1/2)*sin2x
x∈[0,π/2]
0≤2x≤π
所以g(x)的最大值是1/2
当2x=π/2,即x=π/4时取的最大值.