At which points on the curve y = 1 + 30x3 − 3x5 does the tangent line have the largest slope?

Answer 1

At what point on the curve y = 1 + 30x³ - 3x⁵ have the largest slope?

This is the same as asking for the maxima of the derivative

dy/dx = 90 x^2 - 15 x^4

which we can find by locating the zeros of the second derivative

d²y/dx² = 180 x - 60 x^3

These roots are ± √3 and 0. But the next derivative

d³y/dx³ = 180 - 180 x^3

is negative at ± √3 and positive at 0, so the largest slope of the curve is at the points (-√3, 1 - 63√3) and (√3, 1 + 63√3)