How do use the first derivative test to determine the local extrema #36x^2 +24x^2#?
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To use the first derivative test to determine the local extrema of the function f(x) = 36x^2 + 24x^2, follow these steps:

Find the first derivative of the function f'(x) using the power rule. f'(x) = d/dx (36x^2 + 24x) = 72x + 24.

Set f'(x) equal to zero and solve for critical points. 72x + 24 = 0 x = 24/72 x = 1/3.

Evaluate the sign of the first derivative f'(x) in intervals around the critical point. Choose test points within these intervals and plug them into f'(x) to determine the sign. For x < 1/3: Pick x = 1. f'(1) = 72(1) + 24 = 48 (negative) For 1/3 < x < 1/3: Pick x = 0. f'(0) = 72(0) + 24 = 24 (positive) For x > 1/3: Pick x = 1. f'(1) = 72(1) + 24 = 96 (positive).

Analyze the signs of the first derivative to determine the nature of extrema.
 Negative to positive at x = 1/3 indicates a local minimum.
Therefore, the function has a local minimum at x = 1/3.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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