relative maximum
In mathematics, the concept of a relative maximum refers to the highest point on a graph within a specific interval or region
In mathematics, the concept of a relative maximum refers to the highest point on a graph within a specific interval or region. This is different from an absolute maximum, which refers to the overall highest point on the entire graph.
To find a relative maximum, we can follow a few steps:
1. Identify the region or interval over which we want to find the relative maximum. This may be given in the problem statement or determined based on the context of the graph.
2. Determine the critical points within this region. Critical points are the points where the derivative of the function is either zero or undefined. These points may indicate potential relative maximums.
3. Evaluate the function at the critical points and at the endpoints of the interval. This will help us compare the function values and identify the highest point.
4. The point with the highest function value within the given interval is the relative maximum.
Let’s take an example to illustrate this process:
Consider the function f(x) = 2x^3 – 3x^2 – 12x. We want to find the relative maximum of this function within the interval [-2, 3].
Step 1: The given interval is [-2, 3].
Step 2: To find the critical points, we need to differentiate the function. Taking the derivative of f(x), we have:
f'(x) = 6x^2 – 6x – 12.
Now, we need to solve for x when f'(x) = 0.
6x^2 – 6x – 12 = 0.
Factoring or using the quadratic formula, we find that x = -1 and x = 3 are the critical points.
Step 3: We need to evaluate the function at the critical points and at the endpoints of the interval.
f(-2) = 2(-2)^3 – 3(-2)^2 – 12(-2) = -28,
f(3) = 2(3)^3 – 3(3)^2 – 12(3) = -27,
and f(-1) = 2(-1)^3 – 3(-1)^2 – 12(-1) = -5.
Step 4: From the calculations, we see that the function evaluates to a higher value at x = -2 compared to the critical points at x = -1 and x = 3. Therefore, the relative maximum of f(x) within the interval [-2, 3] is f(-2) = -28.
It is important to note that there can be multiple relative maximums within different intervals. To determine the overall absolute maximum, we need to consider all the relative maximums from all the intervals.
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