# For what values of x is #f(x)=4x^3-3x+5# concave or convex?

Convexity and concavity are determined by the sign of the second derivative.

Find the second derivative of the function.

Thus,

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To determine where the function ( f(x) = 4x^3 - 3x + 5 ) is concave or convex, we need to analyze its second derivative, ( f''(x) ).

[ f'(x) = 12x^2 - 3 ] [ f''(x) = 24x ]

The function will be concave upward (convex) where ( f''(x) > 0 ), and concave downward (concave) where ( f''(x) < 0 ).

Setting ( f''(x) > 0 ): [ 24x > 0 ] [ x > 0 ]

Setting ( f''(x) < 0 ): [ 24x < 0 ] [ x < 0 ]

Therefore, ( f(x) = 4x^3 - 3x + 5 ) is concave upward (convex) for ( x > 0 ) and concave downward (concave) for ( x < 0 ).

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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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