What is the largest possible product of two non-negative number whose sum is 1
see other post.
y = 1-x
z = xy = x(1-x)
now just use what you know about parabolas. No calculus needed here.
Of course, you can verify that the max occurs at the vertex...
To find the largest possible product of two non-negative numbers whose sum is 1, we can use the concept of maximizing a function. Let's denote the two numbers as x and y.
1. Write the equation representing the given constraint:
x + y = 1
2. Express one variable in terms of the other.
We can rewrite the equation as x = 1 - y.
3. Formulate the function to maximize.
We want to maximize the product xy. Substituting the value of x from step 2, the function becomes:
f(y) = y(1 - y)
4. Determine the critical points.
To find the critical points, we take the derivative of f(y) with respect to y and set it equal to zero:
f'(y) = 1 - 2y = 0
Solving for y, we get y = 1/2.
5. Check the endpoints.
Since we are dealing with non-negative numbers, we should also evaluate the function at the endpoints of the feasible interval (0 and 1).
f(0) = 0(1 - 0) = 0
f(1) = 1(1 - 1) = 0
The function evaluates to zero at both endpoints.
6. Evaluate the function at the critical points.
f(1/2) = (1/2)(1 - 1/2) = 1/4
7. Compare the values obtained.
Comparing the values at the critical points and endpoints, we can see that the largest value of the function f(y) is 1/4, which occurs when y = 1/2.
8. Find the corresponding x-value.
Using the equation x = 1 - y, we can calculate x = 1 - 1/2 = 1/2.
Therefore, the largest possible product of two non-negative numbers whose sum is 1 is 1/4, and the two numbers are both 1/2.