Is #f(x)=-2x^5-2x^4+5x-45# concave or convex at #x=-2#?

Question

Aurora Alvarado · Accepted Answer

A function (or its graph) can be said to be concave or convex on an interval. This function is convex near #-2#. (In some open interval containing #-2#.)

A necessary and sufficient condition for #f# to be convex on an interval is that #f''(x) >0# for all #x# in the interval. In this case, #f''(x) = -40x^3-24x^2#. So, #f''(-2) = -40(-8)-24(4) <0# #f''(x)# is continuous near #-2#, so #f''(x) <0# for #x# near #-2# and #f# is convex near #-2#. (If you have been given a definition of #f# is "convex at a number, #a#", then I would guess you'll say #f# is convex at #-2#.)

Emma Aldridge · Answer

undefined To determine whether the function $ f(x) = -2x^5 - 2x^4 + 5x - 45 $ is concave or convex at $ x = -2 $, we need to analyze the second derivative of the function at that point.

Taking the first and second derivatives of $ f(x) $:

First derivative:
$$ f'(x) = -10x^4 - 8x^3 + 5 $$

Second derivative:
$$ f''(x) = -40x^3 - 24x^2 $$

Now, evaluate $ f''(-2) $:

$$ f''(-2) = -40(-2)^3 - 24(-2)^2 $$
$$ f''(-2) = -40(-8) - 24(4) $$
$$ f''(-2) = 320 - 96 $$
$$ f''(-2) = 224 $$

Since $ f''(-2) > 0 $, the function is concave upward at $ x = -2 $.