Answer
See the explanation
Work Step by Step
The First Derivative Test is a method used to determine whether a critical point of a function corresponds to a local maximum, local minimum, or neither. Here's how it works:
1. Find the critical points: Compute the first derivative of the function and solve for points where the derivative is zero or undefined. These points are potential candidates for local extrema.
2. Test the sign changes: Around each critical point, examine the sign of the derivative. If the derivative changes sign from positive to negative at a critical point, then it indicates a local maximum. Conversely, if the derivative changes sign from negative to positive at a critical point, then it indicates a local minimum.
3. Verify the concavity: If the derivative doesn't change sign at a critical point, further examine the behavior of the function to determine the type of extremum. Use the second derivative test or examine the concavity to make this determination.
Now, let's illustrate this with examples:
Example 1: \(f(x) = x^3 - 3x^2 - 9x + 5\)
1. Find the critical points: Calculate the first derivative: \(f'(x) = 3x^2 - 6x - 9\). Setting this equal to zero, we find critical points at \(x = -1\) and \(x = 3\).
2. Test the sign changes: Around \(x = -1\), \(f'(-2) = 15 > 0\) and \(f'(0) = -9 < 0\), indicating a local maximum at \(x = -1\). Around \(x = 3\), \(f'(2) = -3 < 0\) and \(f'(4) = 15 > 0\), indicating a local minimum at \(x = 3\).
Example 2: \(g(x) = x^4 - 4x^3 + 6x^2 + 4x - 1\)
1. Find the critical points: Calculate the first derivative: \(g'(x) = 4x^3 - 12x^2 + 12x + 4\). Setting this equal to zero, we find one critical point at \(x = 1\).
2. Test the sign changes: Around \(x = 1\), \(g'(0) = 4 > 0\) and \(g'(2) = 20 > 0\), indicating that \(x = 1\) does not correspond to a local extremum.
In the second example, since there's no sign change around the critical point, we can't determine if it's a local maximum or minimum using the First Derivative Test alone. Additional tests, like the second derivative test or analyzing the behavior of the function around the critical point, would be needed.
