Identifying Extrema


In Exercises 41–52:


a. Identify the function’s local extreme values in the given domain, and say where they occur.


g(x) = x² − 4x + 4, 1 ≤ x < ∞

Verified step by step guidance

1. First, understand that local extrema refer to the local minimum or maximum values of a function within a given domain. For the function g(x) = x² − 4x + 4, we need to find these values within the domain 1 ≤ x < ∞.
2. To find the local extrema, we start by taking the derivative of the function g(x). The derivative, g'(x), will help us find the critical points where the slope of the tangent is zero or undefined. Calculate the derivative: g'(x) = d/dx (x² − 4x + 4).
3. Set the derivative equal to zero to find the critical points: g'(x) = 0. Solve the equation to find the values of x where the slope is zero. This will give us potential points for local extrema.
4. Once you have the critical points, evaluate the second derivative, g''(x), to determine the concavity at these points. If g''(x) > 0, the function has a local minimum at that point. If g''(x) < 0, the function has a local maximum.
5. Finally, check the critical points within the given domain 1 ≤ x < ∞ to ensure they are valid. Evaluate g(x) at these points to find the local extreme values and specify where they occur.

Key Concepts

Local Extrema

Local extrema refer to the points in a function where it reaches a local maximum or minimum value. These points occur where the derivative of the function is zero or undefined, indicating a change in the direction of the slope. Identifying local extrema involves finding these critical points and using tests like the first or second derivative test to determine their nature.

Derivative and Critical Points

The derivative of a function provides the rate of change or slope of the function at any given point. Critical points occur where the derivative is zero or undefined, which are potential locations for local extrema. For the function g(x) = x² − 4x + 4, finding the derivative and setting it to zero helps identify these critical points within the specified domain.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, generally expressed in the form ax² + bx + c. They graph as parabolas, which can open upwards or downwards. The vertex of the parabola represents the extremum of the function. For g(x) = x² − 4x + 4, completing the square or using the vertex formula can help find the vertex, which is crucial for identifying the local extremum.