Calculus Problem

Question

Consider the function f(x)= x^5+7sqrtx.

Let F(x) be the antiderivative of f(x) with F(1) = -4.

Then F(x) =

Answer 1

f(x) = x⁵ + 7x^1/2

F(x) = (1/6)x⁶ + (2/3)7x^3/2 + C

= x⁶/6 + 14x^3/2/3 + C

When x = -1, the function value is -4:

-4 = 1/6 + 14/3 + C

C = -24/6 - 1/6 - 28/6 = -53/6

F(x) = x⁶/6 + 14x^3/2/3 - 53/6

This Desmos graph shows that the point (-1,4) is on the curve for F(x) and that F'(x) matches f(x):

desmos.com/calculator/d5f7bmspab