In the viewing rectangle [-4, 4] by [-20, 20], graph the function f(x) = x^3 - 3x and its secant line through the points (-3, -18) and (3, 18).

Question

In the viewing rectangle [-4, 4] by [-20, 20], graph the function f(x) = x^3 - 3x and its secant line through the points (-3, -18) and (3, 18).

Find the values of the numbers c that satisfy the conclusion of the Mean Value Theorem for the interval [-3, 3].

Answer 1

Hey Chris.

 

The secant line equation for this function will be the first thing you need to figure out:

 

You're given two points, so use the point-slope form:

 

y - y1 = m*(x - x1)

m = (y2 - y1) / (x2 - x1)

(x1, y1) = (-3, -18)

(x2, y2) = (3, 18)

 

m = (18 - -18) / (3 - -3)

m = 36/6

m = 6

 

y - -18 = 6*(x - -3)

y + 18 = 6*(x + 3)

y = 6x

 

Since the slope of your secant line is 6, you need to find the values c such that the slope of the tangent line of f(x) is also 6 in order to satisfy MVT.

 

f(x) = x^3 - 3x

f'(x) = 3x^2 - 3

 

f'(c) = 3*c^2 - 3 = 6

3*c^2 = 9

c^2 = 3

c = sqrt(3), -sqrt(3)

 

Both of these results fit in your interval of [-3, 3], so both results apply.