# How does the first derivative test differ from the second derivative test?

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The first derivative test examines the sign changes of the first derivative of a function to determine the relative extrema. If the first derivative changes sign from positive to negative at a point, it indicates a local maximum, while a change from negative to positive indicates a local minimum.

The second derivative test involves analyzing the concavity of a function to determine the nature of its critical points. If the second derivative is positive at a critical point, it implies the function is concave up, suggesting a local minimum. Conversely, if the second derivative is negative at a critical point, it indicates the function is concave down, implying a local maximum. If the second derivative is zero at a critical point, 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.

- Given that #y= e^-x sinbx#, where #b# is a constant, show that # (d^2y)/(dx^2) + 2dy/dx + ( 1 + b^2) y = 0 # ?
- Is #f(x)=10x^5-7x^4+x-4# concave or convex at #x=-1#?
- How do you find all local maximum and minimum points using the second derivative test given #y=x^5-x#?
- How do you describe the concavity of the graph and find the points of inflection (if any) for #f(x) = x^3 - 3x + 2#?
- Is #f(x)=9x^3+2x^2-2x-2# concave or convex at #x=-1#?

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