Is #f(x)=3/x-2x# concave or convex at #x=9/4#?

Question

Charles Arocha · Accepted Answer

Convex

Find the value of the second derivative at #x=9/4#. If the value is positive, then the function is convex. If it's negative, the function is concave at that point. To find the function's second derivative, recall that #3/x=3x^-1# and then apply the power rule. #f(x)=3x^-1-2x# #f'(x)=-3x^-2-2# #f''(x)=6x^-3=6/x^3# The value of the second derivative at #x=9/4# is #f''(9/4)=6/(9/4)^3# This could be simplified, but it is evidently positive. Thus the function is convex at #x=9/4#.

Christopher Agresta · Answer

undefined To determine whether the function $ f(x) = \frac{3}{x} - 2x $ is concave or convex at $ x = \frac{9}{4} $, we need to find the second derivative and evaluate it at $ x = \frac{9}{4} $.

First, find the first derivative of $ f(x) $, then find the second derivative.

$ f'(x) = -\frac{3}{x^2} - 2 $

Now, find the second derivative:

$ f''(x) = \frac{6}{x^3} $

At $ x = \frac{9}{4} $, $ f''\left(\frac{9}{4}\right) = \frac{6}{\left(\frac{9}{4}\right)^3} $

Calculate $ \frac{9}{4} $ cubed and then calculate $ \frac{6}{\left(\frac{9}{4}\right)^3} $.

If $ f''\left(\frac{9}{4}\right) > 0 $, the function is concave up at $ x = \frac{9}{4} $.

If $ f''\left(\frac{9}{4}\right) < 0 $, the function is concave down at $ x = \frac{9}{4} $.

If $ f''\left(\frac{9}{4}\right) = 0 $, the test is inconclusive, and we may need additional information to determine concavity.