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Is #f(x)=4x^5-12x^4+5x^2+2x+2# concave or convex at #x=-5#?

Answer 1

Concave

We use the second derivative to determine the curvature of a function. It is concave if the second derivative is less than zero and convex if the second derivative is greater than zero.

#f(x)=4x^5−12x^4+5x^2+2x+2#

#f'(x)=20x^4−48x^3+10x+2#

#f''(x)=80x^3−144x^2+10#

#f''(-5) = 80(-5)^3−144(-5)^2+10#

#f''(-5) = -10000 − 3600 + 10 = -13590#

Definitely negative, so it is concave.

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Answer 2

Elijah Abram

To determine if the function ( f(x) = 4x^5 - 12x^4 + 5x^2 + 2x + 2 ) is concave or convex at ( x = -5 ), you need to analyze the second derivative of the function at that point.

Find the second derivative of ( f(x) ), denoted as ( f''(x) ).
Evaluate ( f''(-5) ).
If ( f''(-5) > 0 ), the function is concave upward (convex) at ( x = -5 ).
If ( f''(-5) < 0 ), the function is concave downward (concave) at ( x = -5 ).
If ( f''(-5) = 0 ), the test is inconclusive.

You can find the second derivative by taking the derivative of the first derivative of the function ( f(x) ). Once you have ( f''(x) ), substitute ( x = -5 ) to determine the concavity at that point.

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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.