# How do you determine the concavity of a quadratic function?

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In any function, if the second derivative is positive, the function is concave up. If the second derivative is negative, the function is concave down.

This can be shown graphically:

graph{6x^2+3x-5 [-18.5, 17.54, -10.35, 7.68]}

graph{-1/2x^2-7x+1 [-64.2, 52.83, -24.88, 33.7]}

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To determine the concavity of a quadratic function, you need to analyze its second derivative.

- Compute the first derivative of the quadratic function.
- Then, compute the second derivative of the function.
- If the second derivative is positive for all x-values in the domain, the function is concave up.
- If the second derivative is negative for all x-values in the domain, the function is concave down.
- If the second derivative changes sign at a point within the domain, the function changes concavity at that point.

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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.

- Is #f(x)=xcosx# concave or convex at #x=pi/2#?
- What intervals is #f(x) = (5x^2)/(x^2 + 4)# concave up/down?
- For what values of x is #f(x)= e^x/(5x^2 +1# concave or convex?
- How do you find local maximum value of f using the first and second derivative tests: #f(x) = x^5 - 5x + 5#?
- How do you find the points of Inflection of #f(x)=2x(x-4)^3#?

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