Is #f(x)=9x^3+2x^2-2x-2# concave or convex at #x=-1#?

Question

Ezekiel Alexander · Accepted Answer

Concave (this is also called concave down).

The concavity or convexity of a function are determined by the sign of the second derivative. Finding the second derivative of the function is a simple application of the power rule. #f(x)=9x^3+2x^2-2x-2# #f'(x)=27x^2+4x-2# #f''(x)=54x+4# Find the sign of the second derivative at #x=-1#: #f''(-1)=-54+4=-50# Since this is #<0#, the function is concave at #x=-1#. Concavity means that the function resembles the #nn# shape. We can check the graph of #f(x)#: graph{9x^3+2x^2-2x-2 [-3, 3, -15, 15]}

Ezra Adams · Answer

undefined To determine the concavity of $ f(x) = 9x^3 + 2x^2 - 2x - 2 $ at $ x = -1 $, we need to examine the second derivative of the function.

First, find the first derivative:

$ f'(x) = 27x^2 + 4x - 2 $

Then, find the second derivative:

$ f''(x) = 54x + 4 $

Now, evaluate the second derivative at $ x = -1 $:

$ f''(-1) = 54(-1) + 4 = -54 + 4 = -50 $

Since the second derivative $ f''(-1) $ is negative, the function is concave downward at $ x = -1 $.