# How do you know when to use the first derivative test and the second derivative test?

You use the First Derivative Test when determining the relative extrema of a function. It involves analyzing the sign changes of the first derivative in the vicinity of critical points.

You use the Second Derivative Test when determining the nature of critical points, specifically whether they correspond to local maxima, local minima, or points of inflection. This test utilizes the concavity information provided by the sign of the second derivative near critical points.

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Unless a problem specifically requests otherwise, I would always use the first derivative test because it only requires taking the derivative once and always yields a conclusion; the second derivative test is useful in situations where finding the second derivative is straightforward, but it is ineffective when the second derivative is zero at a critical value.

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The first derivative test is used to determine the local extrema (maximum and minimum points) of a function. It examines the sign changes of the derivative around critical points. If the derivative changes sign from positive to negative at a critical point, it indicates a local maximum, and if it changes from negative to positive, it indicates a local minimum. If the derivative does not change sign at a critical point, the test is inconclusive.

The second derivative test is used when the first derivative test is inconclusive or difficult to apply. It examines the concavity of the function near critical points. If the second derivative is positive at a critical point, it indicates that the function is concave upward, implying a local minimum. If the second derivative is negative at a critical point, it indicates that the function is concave downward, implying a local maximum. If the second derivative is zero, the test is inconclusive.

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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.

- For what values of x is #f(x)=(x-1)(x-7)(x-1)# concave or convex?
- If #(x^2+1) / (x^2-2)#, what are the points of inflection, concavity and critical points?
- How do you determine the intervals where #f(x)=3x^2-x^3/3# is concave up or down?
- How do you use the first and second derivatives to sketch # y = x + (1-x)^(1/2)#?
- How do you find all points of inflection given #y=-x^3/(x^2-1)#?

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