Find a function f such that f '(x) = 5x^3 and the line 5x + y = 0 is tangent to the graph of f.

Question

Find a function f such that f '(x) = 5x³ and the line 5x + y = 0 is tangent to the graph of f.

f(x)=_____

Answer 1

We integrate to get f(x) = ∫5x³dx = 5x⁴/4 + C.

Because tangent line is y = - 5x, then f'(x) = 5x³ = - 5, x³ = -1 and x = -1 is x-coordinate of tangent point.

Then y-coordinate of tangent point is f(-1) = 5(-1)⁴/4 + C = 5/4 + C.

Then equation of tangent line is y - 5/4 - C = - 5(x + 1), y = - 5x - 15/4 + C.

So -15/4 + C = 0, C = 15/4 and f(x) = 5x⁴/4 + 15/4