Question

c be the curve of the equation y=3x^5+10x^4-10x^3+60x^2+25x-5. among all points on the curve over the interval [-1,3], find the coordinates of the highest slope and lowest slope

Answer 1

the slope is y '

y ' = 15x^4 + 40x^3 - 30x^2 + 120x + 25

the max/min happens when y '' = 0
y '' = 60x^3 + 120x^2 - 60x + 120

so setting that to zero
60x^3 + 120x^2 - 60x + 120 = 0
x^3 + 2x^2 - x + 2 = 0
x^2(x+2) - (x+2) =
(x+2)(x^2-1) = 0
x = ± 1 or x = -2

so within our given domain, we use x = ±1
if x=1 , y ' = 15 + 40 - 30 + 120 + 25 = 170

I will let you finish it.