For what values of x is #f(x)= -x^3+3x^2-2x+2 # concave or convex?

1 Answer
Nov 19, 2017

Concave (Convex) Up on the interval #( -oo,1 )#
Concave (Convex) Down on the interval #( 1, oo )#

Explanation:

We are given the function #f(x) = -x^3 + 3x^2 - 2x + 2#

#color(red)(Step.1)#

Find the First Derivative

#f'(x) = -3x^2 + 6x -2#

#color(red)(Step.2)#

Find the Second Derivative

#f''(x) = -6x + 6#

#color(red)(Step.3)#

Next, set

#f''(x) = -6x + 6 = 0#

Simplifying, we get #x = 1#

#color(red)(Step.4)#

Then, we consider a number larger than 1 and a number smaller than 1 and substitute the values in our Second Derivative.

If the number is Greater than 1, our #f''(x) = -6x + 6"# will yield a "Negative" number.

If the number is Less than than 1, our #f''(x) = -6x + 6"# will yield a "Positive" number.

Hence, we observe that #f(x)# is "Concave Up" on the interval #(-oo, 1)# and "Concave Down" on the interval #(1, oo)#

Refer to the Number Line as shown below:

                                                    1

:..........................................................*..................................................................:

             Positive                                                  Negative