Let f(a) = S02|x(x-a)| dx for 0< a <

(1) Find the function f(a).

(2) Find the minimum of f(a).

Thank you very much for your help!

Originally Posted by kokoro123

Let f(a) = S02|x(x-a)| dx for 0< a <

(1) Find the function f(a).

(2) Find the minimum of f(a).
The integral becomes $\int_0^a {x(a - x)dx} + \int_a^2 {x(x - a)dx}$

Originally Posted by kokoro123
(1) Find the function f(a).
Split it up:

$\int_0^2\left|x(x-a)\right|\,dx$

$=\int_0^a\left|x(x-a)\right|\,dx + \int_a^2\left|x(x-a)\right|\,dx$

$=\int_0^a\left[-x(x-a)\right]\,dx + \int_a^2x(x-a)\,dx$

Originally Posted by kokoro123
(2) Find the minimum of f(a).
Use the Fundamental Theorem of Calculus to differentiate $f$. Locate critical values, etc.