Find two positive numbers with product 400 and whose sum is a minimum. Enter your answers in increasing order. First number: Number Second Number: Number
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.
![### Problem Statement
Find two positive numbers with a product of 400 and whose sum is a minimum.
### Instructions
Enter your answers in increasing order.
**First number:** [Number input field]
**Second Number:** [Number input field]
### Explanation
To solve this problem, follow these steps:
1. **Formulate the Equations:**
Let the two numbers be \( x \) and \( y \). The key equations are:
\[
x \cdot y = 400
\]
\[
x + y = \text{minimum}
\]
2. **Express One Variable in Terms of the Other:**
From \( x \cdot y = 400 \), we get:
\[
y = \frac{400}{x}
\]
3. **Form the Sum Equation:**
Substitute \( y \) in the sum equation \( x + y \):
\[
x + \frac{400}{x} = \text{minimum}
\]
4. **Find the Minimum Value:**
To find the minimum value, we can take the derivative of \( S = x + \frac{400}{x} \) with respect to \( x \), then set the derivative to 0 and solve for \( x \).
5. **Evaluate the Corresponding \( y \):**
Once you have the value for \( x \), substitute it back into the equation \( y = \frac{400}{x} \) to find \( y \).
6. **Enter in Increasing Order:**
Input the values of the numbers in the provided fields in increasing order.
By following these steps, you should be able to determine the two positive numbers whose product is 400 and whose sum is minimized.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F471cb0b1-24a7-4f47-a754-0c03ff5b6852%2F26a01e49-0ebd-4083-82df-2eae26f2936b%2Fcv4zndx_processed.png&w=3840&q=75)
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