Home > Calculus > Analyzing Concavity of a Function

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

Answer 1

Aurora Alvarado

Convex.

You can tell if a function is concave or convex by the sign of its second derivative:

To find the second derivative, apply the power rule to each term twice.

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

#f'(x)=-10x^4-6x^2+6x-1#

#f''(x)=-40x^3-12x+6#

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

#f''(-1)=-40(-1)^3-12(-1)+6#

This mostly becomes a test of keeping track of your positives and negatives.

#f''(-1)=-40(-1)+12+6=40+18=58#

Since this is #>0#, the function is convex at #x=-1#. Convexity on a graph is characterized by a #uu# shape.

We can check the graph of the original function:

graph{-2x^5-2x^3+3x^2-x+3 [-2.5, 2, -30, 30]}

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Answer 2

Scarlett Allison

The function f(x) = -2x^5 - 2x^3 + 3x^2 - x + 3 is concave at x = -1.

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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.