Is #f(x)=-2x^5+7x^2-6x+3# concave or convex at #x=-4#?

1 Answer
mason m
Feb 25, 2016

Convex.

Explanation:

To test a function's concavity or convexity, find the value of the second derivative of the function at a point:

  • If #f''(-4)<0#, then #f(x)# is concave at #x=-4#.
  • If #f''(-4)>0#, then #f(x)# is convex at #x=-4#.

To find the function's second derivative, use the power rule twice:

#f(x)=-2x^5+7x^2-6x+3#

#f'(x)=-10x^4+14x-6#

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

The value of the second derivative at #x=-4# is

#f''(-4)=-40(-4)^3+14=-40(-64)+14=2574#

Since this is #>0#, the function is convex at #x=-4#.

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