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)?

f'= 3x2 + 8x

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

so u replace x by 5 in fx and u substract

but how would you get f(5)?

Use 5 for x

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

?? 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?

To find f'(5) - f(5), we need to first differentiate the equation f(x)=x^3+4x^2+∫(from 0 to x)(x−t)f′(t) dt with respect to x.

Let's differentiate the equation step by step:

Differentiating f(x), we get:

f'(x) = 3x^2 + 8x + ∫(from 0 to x)(x−t)f′(t) dt

Next, we differentiate the integral term using the Leibniz integration rule:

d/dx ∫(from 0 to x)(x−t)f′(t) dt = (x−x)f′(x) + ∫(from 0 to x)(-f′(t)) dt

The first term on the right-hand side will evaluate to zero, so we are left with:

f'(x) = 3x^2 + 8x - ∫(from 0 to x)f′(t) dt

Now, let's evaluate f'(5) - f(5):

f'(5) - f(5) = (3*5^2 + 8*5 - ∫(from 0 to 5)f′(t) dt) - (5^3 + 4*5^2 + ∫(from 0 to 5)(5−t)f′(t) dt)

Simplifying further:

f'(5) - f(5) = (75 + 40 - ∫(from 0 to 5)f′(t) dt) - (125 + 100 + ∫(from 0 to 5)(5−t)f′(t) dt)

Combining like terms:

f'(5) - f(5) = (115 - ∫(from 0 to 5)f′(t) dt) - (225 + ∫(from 0 to 5)(5−t)f′(t) dt)

Now, we need to integrate f'(t) and substitute the limits of integration to simplify further. However, without more information about the specific function f(x) or f'(x), we cannot determine the exact values of the integral terms.