Question

find the tangents to the curve y=2x^3 +3x at the points where the slope is 9. what is the smallest slope on the curve? at what value of x does the curve have this slope? please show all steps!

Answer 1

differentiate the function to get the slope of the function at all points
y' = 6x^2 +3
so 9 = 6x^2 +3
so x = +-1
if x is 1 put it into the original y = 2x^3 +3x
y = 2 * 1^3 +3*1
y = 2 +3
y = 5
point is (1,5)
sub -1
y = 2 * (-1)^3 +3(-1)
y = -2 -3
y = -5
so point is (-1,5)

Answer 2

second part. y' (the slope) is at a min i.e 0 when x = 0
so ans 0 and 0 !!!