What are the values for x if the graph of f(x) = (x + 1)(x - 6)^2 ...is concave upward?

Question

hmmm... I got

-8/3 < x < 2

4/3 < x < 6 for upward

Answer 1

To determine if a graph is concave upward, you need to take the double derrivative. A positive number indicates a concave upward.

foil out the equation to get:

(x+1)(x^2 - 12x + 36)

x^3 - 11x^2 +24x +36

now take the first derivative:

df/dx = 3x^2 - 22x + 24

now take the double derrivative

d(df/dx) = 6x - 22

the value is concave up when d(df/dx) is > 0

so, X < 22/6 is concave down, and X> 22/6 is concave up

hope that helps - do your homework yourself now :-)

Answer 2

J Russell,
Without solving, since that's already been done...
The highest power in your equation is 3. Which means the highest power in your second derivative is 1; it's a straight line. The straight line can only cross the x-axis at one point; so you can't get more than one interval as a response, and that interval must be a ray: (∞,a) or (a,∞).