Question

given the curve y= 1/3 x3 (cubed) - 9x - 7

a) evaluate dy/.dx

b)for what values of x is dy/dx = 0

c)hence, state the values of x at which the curve possesses stationary points. (note not required to evaluate corresponding y values)

d)determine the nature of the stationary points of the curve.

Answer 1

given the curve y= 1/3 x³ - 9x - 7
&
a) evaluate dy/dx

dy/dx = (1/3)*3 x² - 9 - 0 = x² - 9.

b)for what values of x is dy/dx = 0

x² - 9 = 0
(x-3)(x+3) = 0

x = ±3 - zeros of dy/dx.
&
c)hence, state the values of x at which the curve possesses stationary points. (note not required to evaluate corresponding y values)

x = ±3 - stationary points.
&
d)determine the nature of the stationary points of the curve.

dy/dx > 0, x<-3;
dy/dx < 0, -3<x<3;
dy/dx > 0, x>3.

So, x=3 and x=-3 are local maximum and minimum points respectively.