f " (x) = X^2 + X -1, f ' (0)=5, f (0)=10
First we need to determine f'(x) by calculating the integral of f''(x)
We know that:
f'(x) = intg f''(x)
==> f'(x) = intg (x^2 + x -1) dx
= x^3/3 + x^2/2 - x + C
==> f'(x) = (1/3)x^2 + (1/2)x^2 - x + C
Given that:
f'(0) = 5
==> f'(x) = (1/3)x^3 + (1/2)x^2 - x + 5
Now let us calculate f(x)"
We know that:
f(x) = intg f'(x)
= intg ((1/3)x^3 + (1/2)x^2 - x + 5) dx
= (1/3)x^4/4 + (1/2)x^3/3 - x^2/2 + 5x + C
= (1/12)x^4 + (1/6)x^3 - (1/2)x^2 + 5x + C
Given that:
f(0) = 10 ==> c = 10
==> f(x) = (1/12)x^4 + (1/6)x^3 - (1/2)x^2 + 5x + 10