# Is #f(x)=-2x^5-2x^4+5x-45# concave or convex at #x=-2#?

A function (or its graph) can be said to be concave or convex on an interval. This function is convex near

In this case,

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To determine whether the function ( f(x) = -2x^5 - 2x^4 + 5x - 45 ) is concave or convex at ( x = -2 ), we need to analyze the second derivative of the function at that point.

Taking the first and second derivatives of ( f(x) ):

First derivative: [ f'(x) = -10x^4 - 8x^3 + 5 ]

Second derivative: [ f''(x) = -40x^3 - 24x^2 ]

Now, evaluate ( f''(-2) ):

[ f''(-2) = -40(-2)^3 - 24(-2)^2 ] [ f''(-2) = -40(-8) - 24(4) ] [ f''(-2) = 320 - 96 ] [ f''(-2) = 224 ]

Since ( f''(-2) > 0 ), the function is concave upward at ( x = -2 ).

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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.

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.

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.

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