Question

At which x values does the curve y=1+60x^3-10x^5 does the tangent line have the largest slope. List the x values in increasing order and enter n in the second box if the max occurs at only one x value

Answer 1

the slope is given by dy/dx
dy/dx = 180x^2 - 50x^4

This will have a max (or min) if its derivative is zero, that is,
360x - 200x^3 = 0
x(360 - 200x^2) = 0
x = 0 or x = ±360/200 = ±9/5

when x = 0, dy/dx = 180(0) - 50(0) = 0
when x = 9/5 , dy/dx = 180(81/25) - 50(6561/625)
= 1458/25
when x = -9/5 , dy/dx = 180(-81/25) - 50(-6561/625)
= -1458/25

So,when x = -9/5 the slope is a minimum
when x = 9/5 the slope is a maximum

check my arithmetic.