You need to use the mean value thorem to verify the given inequality, such that:
`int_a^b f(x)dx = (b-a)*f(c), c in (a,b)`
Replacing `x^2 - 4x + 4` for `f(x)` and 0 for a, 4 for b, yields:
`int_0^4 (x^2 - 4x + 4) dx= (4-0)f(c) = 4f(c)`
You need to prove that `int_0^4 (x^2 - 4x + 4) dx >= 0,` hence, according to mean value theorem, `4f(c)>=0.`
You need to check the monotony of the function f(x), hence, you need to find the derivative using the chain rule and to verify if it is positive or negative on interval (0,4), such that:
`f(x) = (x^2 - 4x + 4) => f(x) = (x-2)^2 => f'(x) = 2(x-2)*(x-2)'`
`f'(x) = 2(x-2) `
You need to notice that f(x) decreases on (0,2) and it increases on (2,4).
Hence, for c in (0,2) yields:
`0<c<2 => f(0)>f(c)>f(2)`
You need to evalute f(0) and f(2), such that:
`f(0) = (0-2)^2 = 4`
`f(2) = (2-2)^2 = 0`
Hence, replacing the found values in inequality `f(0)>f(c)>f(2)` , yields:
You need to obtain 4f(c), hence, multiplying by positive 4 both sides yields:
But `4f(c) = int_0^4 (x^2 - 4x + 4) dx` and 4f(c)>0, hence `int_0^4 (x^2 - 4x + 4) dx >0.`
Hence, using mean value theorem, yields that the inequality `int_0^4 (x^2 - 4x + 4) dx >0` holds. Notice that it is no need to evaluate the integral to prove the inequality.
`int_0^4(x^2-4x+4)dx = int_0^4(x-2)^2dx`
Here `(x-2)^2 gt=0` So the given integral is greater than or equal to 0.