Antiderivatives

Question

find f.

f''(x)= 20x^3 + 12x^2 + 6 f(0)=7 f(1)= 4

f'''(x)=cosx, f(0)=9 f'(0)=3 f''(0)=9

Answer 1

I'll get you started on the first one; then you should be able to finish it and do the 2nd one.

Integrate f"(x)"

f'(x)=5x⁴+4x³+6x+C₁

Integrate again

f(x)=x⁵+x⁴+3x²+C₁x+C₂

Now plug in the values for f(0) and f(1) to evaluate the 2 constants.

The second one goes the same way...but it is easier

If f(x) = -cos x

f'(x)=sin x

and f"(x) =cos x

f'(x)=sin x + c₁

and f(x) = -cos x + C₁x + C₂

again plug in the values given to get C₁ and C₂