Given [tex]f(x)=x^a(7-x)^b[/tex] on the interval 0 ≤ x ≤ 7, for maximum value, f'(x) = 0.
[tex]f'(x)=0 \\ \\ \Rightarrow -bx^{a}(7-x)^{b-1}+ax^{a-1}(7-x)^b=0 \\ \\ \Rightarrow ax^{a-1}(7-x)^b=bx^{a}(7-x)^{b-1} \\ \\ \Rightarrow a(7-x)=bx \\ \\ \Rightarrow (a+b)x=7a \\ \\ \Rightarrow x= \frac{7a}{a+b} [/tex]
