Find the antiderivative F of f that satisfies the given condition. Check your answer by comparing the graphs of f and F. f(x) = 2ex − 8x, F(0) = 3

Question

Find the antiderivative F of f that satisfies the given condition. Check your answer by comparing the graphs of f and F.

f(x) = 2e^x − 8x,    F(0) = 3

Answer 1

F(x) = ∫f(x)dx

F(x) = ∫(2e^x − 8x)dx

= 2e^x − 4x² + C

F(0) =3 = 2e⁰ − 4(0)² + C

3 = 2(1) + C

C = 1

Therefore:

F(x) = 2e^x − 4x² + 1

See the graph in this link:

https://www.desmos.com/calculator/i6bflm2arh

(1)You will see that the x-coordinate of local extrema of F(x) matches that of f(x)'s x-intercepts.

(2) From local maximum to local minimum of F(x), the graph of f(x) is below the x-axis.