How do use the first derivative test to determine the local extrema #x^22x3#?
There is a minima of f(x) at x= 1
Since f(x) is defined for all values of x, the only critical point is given by 2x2=0, or, x=1
At x=1, the slope is 0
At any point in #(1, +oo), say x=2, f'(x) would be 42=2. This means the slope of the curve is positive.
The above analysis of the slope of f(x) , shows that the slope of the curve changes from negative, to the left of x=1, to positive to its right. This implies that there is a minima at x=1
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To use the first derivative test to determine the local extrema of the function (f(x) = x^2  2x  3):
 Find the first derivative of the function (f'(x)).
 Determine the critical points by solving (f'(x) = 0).
 Determine the sign of (f'(x)) in intervals between the critical points.
 Use the signs of (f'(x)) to determine the behavior of the function and identify local extrema.
Let's proceed with the steps:

Find the first derivative: [f'(x) = 2x  2]

Determine the critical points: [f'(x) = 0 \implies 2x  2 = 0 \implies x = 1]

Determine the sign of (f'(x)) in intervals:
 When (x < 1), choose (x = 0), then (f'(0) = 2), which means (f'(x) < 0).
 When (x > 1), choose (x = 2), then (f'(2) = 2), which means (f'(x) > 0).

Use the signs of (f'(x)):
 (f'(x) < 0) to the left of (x = 1) indicates a decreasing function.
 (f'(x) > 0) to the right of (x = 1) indicates an increasing function.
Hence, by the first derivative test, there is a local minimum at (x = 1).
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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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