Question

If a and b are positive numbers, find the maximum value of f(x) = xa(7 â x)b on the interval 0 ⤠x ⤠7.

Answer 1

Given $f(x)=x^a(7-x)^b$ on the interval 0 ≤ x ≤ 7, for maximum value, f'(x) = 0.

$f'(x)=0 \\ \\ \Rightarrow -bx^{a}(7-x)^{b-1}+ax^{a-1}(7-x)^b=0 \\ \\ \Rightarrow ax^{a-1}(7-x)^b=bx^{a}(7-x)^{b-1} \\ \\ \Rightarrow a(7-x)=bx \\ \\ \Rightarrow (a+b)x=7a \\ \\ \Rightarrow x= \frac{7a}{a+b}$