Is #f(x)=-2x^5+7x^2+8x-13# concave or convex at #x=-4#?

1 Answer
Dec 4, 2015

Since #f''(-4) > 0#, #f(x)# is convex at #x = -4#.

Explanation:

Let's compute the second derivative:

#f(x) color(white)(ii) = -2x^5 + 7x^2 + 8x - 13#
#f'(x) color(white)(i) = -10 x^4 + 14 x + 8#
#f''(x) = -40 x^3 + 14#

Now, let's evaluate the second derivative at #x = -4# and check if #f''(-4)# is negative or positive:

#f''(-4) = -40 * (-4)^3 + 14#

Even without computing this value, you can see that #(-4)^3# is negative and #-40# is also negative. A negative value multiplied with a positive value is positive. #14# is positive as well.

So we can conclude that

#f''(-4) > 0#

This means that #f(x)# is convex at #x = -4#.