Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
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**Finding Absolute Extrema on a Closed Interval**
In calculus, finding the absolute extrema (both minimum and maximum) on a closed interval involves evaluating the function at both the critical points and the endpoints of the interval.
Let's consider the function given:
\[ g(x) = 5x^2 - 20x \]
We are asked to find the absolute extrema on the closed interval \([0, 5]\).
**Steps to Find Absolute Extrema:**
1. **Find the derivative of the function** to identify critical points:
\[ g'(x) = \frac{d}{dx}(5x^2 - 20x) = 10x - 20 \]
2. **Set the derivative equal to zero** and solve for x:
\[ 10x - 20 = 0 \implies x = 2 \]
3. **Evaluate the function** at the critical point and the endpoints of the interval:
- \( g(0) = 5(0)^2 - 20(0) = 0 \)
- \( g(2) = 5(2)^2 - 20(2) = 20 - 40 = -20 \)
- \( g(5) = 5(5)^2 - 20(5) = 125 - 100 = 25 \)
4. **Identify the absolute minimum and maximum values**:
- The minimum value on \([0, 5]\) occurs at \( (2, -20) \).
- The maximum value on \([0, 5]\) occurs at \( (5, 25) \).
**Conclusion:**
The absolute extrema of the function \( g(x) = 5x^2 - 20x \) on the closed interval \([0, 5]\) are as follows:
- **Minimum**: \( (x, y) = (2, -20) \)
- **Maximum**: \( (x, y) = (5, 25) \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F52a0c9f6-ada1-431f-830a-3fbdcbca3339%2F05ee95b4-b596-43c9-bb74-6567fc9f08ef%2Fvnbm6lp_processed.jpeg&w=3840&q=75)
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