Home > Calculus > Analyzing Concavity of a Function

Is #f(x)=4x^5-2x^4-9x^3-2x^2-6x# concave or convex at #x=-1#?

Answer 1

Charles Arocha

Concave (also called "concave down").

Concavity and convexity are determined by the sign of the second derivative:

Find the second derivative:

#f(x)=4x^5-2x^4-9x^3-2x^2-6x# #f'(x)=20x^4-8x^3-27x^2-4x-6# #f''(x)=80x^3-24x^2-54x-4#

Find the sign of the second derivative when #x=-1#:

#f''(-1)=80(-1)^3-24(-1)^2-54(-1)-4#

#=80(-1)-24(1)+54-4=-80-24+50=-54#

Since #f''(-1)<0#, the function is concave at #x=-1#. This means that it will resemble the #nn# shape. We can check a graph of #f(x)#:

graph{4x^5-2x^4-9x^3-2x^2-6x [-5, 5, -26.45, 19.8]}

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.