First Derivative Test: f(x) has a local Maximum at x=c when f'(x)
To determine whether a function f(x) has a local maximum at x = c, we can use the first derivative test
To determine whether a function f(x) has a local maximum at x = c, we can use the first derivative test. The first derivative test is based on the behavior of the derivative of a function around a particular value.
According to the first derivative test, if the derivative of a function changes sign from positive to negative at a particular value of x, then the function has a local maximum at that point.
So, to determine if f(x) has a local maximum at x = c, we need to evaluate the derivative of f(x), denoted as f'(x), and check its behavior around x = c.
1. Find the derivative of the function f(x) by taking the derivative with respect to x. This will give us f'(x).
2. Evaluate f'(x) at the value x = c.
3. Check the sign of f'(x) around the value x = c. Specifically, check whether f'(x) changes sign from positive to negative as x approaches c from the left.
a. If f'(x) changes sign from positive to negative as x approaches c from the left, then f(x) has a local maximum at x = c.
b. If f'(x) does not change sign or changes sign in a different way, then f(x) does not have a local maximum at x = c.
It’s important to note that the first derivative test does not guarantee the presence of a local maximum, only indicates the possibility of it. To confirm the existence of a local maximum, further analysis may be required, such as evaluating the second derivative at x = c.
Remember, the first derivative test is used specifically for determining local extrema, which includes both maximum and minimum points.
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