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

The answer is

We calculate the first derivative and we build a sign chart

Now we construct the sign chart

Therefore,

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To determine the concavity of ( f(x) = 4x^5 - 5x^4 ), you need to find the second derivative of ( f(x) ) and then examine its sign. If the second derivative is positive, the function is concave up (convex), and if it's negative, the function is concave down. The second derivative of ( f(x) ) is ( f''(x) = 80x^3 - 60x^2 ). To find the values of ( x ) for which the function is concave or convex, set ( f''(x) ) equal to zero and solve for ( x ). Then, test the intervals between the critical points to determine the sign of ( f''(x) ).

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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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