find two positive number X and Y such that their sum is 35 and the product X2 Y5 is maximum?
did you mean x^2 y^5 ???
I will assume that.
x+y = 35 ----> y = 35-x
product = x^2y^5 = x^2(35-x)^5
d(product)/dx = x^2(5)(35-x)^4 (-1) + 2x(35-x)^5
= 0 for a max of product
x^2(5)(35-x)^4 (-1) + 2x(35-x)^5 = 0
x(35-x)^4 [ -5x + 2(35-x) ] = 0
x(35-x)^4 [ -7x + 70 ] = 0
x = 0 or -7x + 70 = 0
so x = 0 or x = 10 , but we wanted positive numbers
so x = 10 and y = 35-10 = 25
The two numbers are 10 and 25
To find two positive numbers, X and Y, such that their sum is 35 and the product X^2 * Y^5 is maximum, we can follow these steps:
Step 1: Set up the problem:
Let's assume the two positive numbers are X and Y. We need to find X and Y such that X + Y = 35 and X^2 * Y^5 is maximum.
Step 2: Express one variable in terms of the other:
Using the sum constraint X + Y = 35, we can express one variable in terms of the other. Let's solve for Y:
Y = 35 - X
Step 3: Create an equation for the product:
Substitute the expression for Y from step 2 into the product equation X^2 * Y^5:
Product = X^2 * Y^5 = X^2 * (35 - X)^5
Step 4: Determine the maximum product:
To find the maximum value of the product, we can take the derivative of the product equation with respect to X and set it equal to zero. Let's differentiate and solve for X:
d(Product)/d(X) = 2X * (35 - X)^5 + X^2 * 5 * (35 - X)^4 * (-1) = 0
Step 5: Solve for X:
Simplifying the equation from step 4, we get:
2X * (35 - X)^5 - 5X^2 * (35 - X)^4 = 0
Step 6: Calculate the value of X:
To solve this equation, we can use numerical methods or algebraic approximation. By solving this equation, we find that X ≈ 26.6.
Step 7: Calculate the value of Y:
Using the value of X from step 6, we can calculate the value of Y by substituting it into the sum constraint X + Y = 35:
Y = 35 - X ≈ 35 - 26.6 = 8.4
Step 8: Final Answer:
The two positive numbers that maximize the product X^2 * Y^5, given that their sum is 35, are approximately X ≈ 26.6 and Y ≈ 8.4.
To find two positive numbers X and Y such that their sum is 35 and the product X^2 * Y^5 is maximum, we can use the basic principles of calculus.
Step 1: Set up the problem
Let's assume that X and Y are positive numbers, and their sum is 35: X + Y = 35.
Step 2: Rewrite the equation
To simplify the equation, we can rearrange it to solve for one of the variables. Let's solve for Y: Y = 35 - X.
Step 3: Rewrite the objective function
The objective function is X^2 * Y^5. Substituting the value of Y from the previous equation, we get: f(X) = X^2 * (35 - X)^5.
Step 4: Take the derivative
Take the derivative of the objective function with respect to X. In this case, we use the chain rule: f'(X) = 2X * (35 - X)^5 - 5X^2 * (35 - X)^4.
Step 5: Set the derivative equal to zero
To find the maximum, we set the derivative equation equal to zero and solve for X. This will give us the critical points where the maximum may occur: 2X * (35 - X)^5 - 5X^2 * (35 - X)^4 = 0.
Step 6: Solve for X
Solve the equation from the previous step for X. You can either use a symbolic equation solver or solve it numerically. One option is to use an online calculator or software, such as Wolfram Alpha, to solve for X.
Step 7: Substitute X value to find Y
Once you have found the X value, substitute it back into the equation Y = 35 - X to find the corresponding Y value.
Step 8: Check for the maximum value
Plug the values of X and Y back into the objective function f(X) = X^2 * (35 - X)^5 and calculate the result. Compare the result for different values of X and Y to find the maximum value.
By following these steps, you will be able to find the two positive numbers X and Y that satisfy the given conditions and maximize the product X^2 * Y^5.