Question

f(x) and f′(x) are continuous, differentiable functions that satisfy

f(x)=x^3+4x^2+∫(from 0 to x)(x−t)f′(t) dt.

What is f′(5)−f(5)?

Answer 1

f'= 3x2 + 8x
f'(5)= 3(25) + 8 (5)=
=75+40=115

so u replace x by 5 in fx and u substract

Answer 2

but how would you get f(5)?

Answer 3

Use 5 for x
5^3 + 4(5)^2 = 125 + 100 = 225

Answer 4

?? if we use f'(x) = 3x^2 + 8x, then that means

f(x) = x^3 + 4x^2 + ∫[0,x] (x-t)(3t^2+8t)
= 1/4 x^4 + 7/3 x^3 + 4x^2

but then f'(x) = x^3 + 7x^2 + 8x

Am I missing something?