Can someone walk me through the following questions: Partial Derivatives
f(x,y) = x^7 -8y -3 find x and y
f(x,y) = 7 / 5x + 4y, find x and y
Sure! Let's go through each question step-by-step:
Question 1: Find x and y in the equation f(x,y) = x^7 - 8y - 3.
To find x and y in this equation, we need to set up partial derivatives. The partial derivative with respect to x (∂f/∂x) will help us find x, and the partial derivative with respect to y (∂f/∂y) will help us find y.
Step 1: Calculate the partial derivative with respect to x (∂f/∂x):
To take the partial derivative with respect to x, treat y as a constant and differentiate x^7. The derivative of x^7 is 7x^6. Since there are no x-terms in the equation except x^7, the overall partial derivative becomes 7x^6.
Step 2: Set the partial derivative with respect to x (∂f/∂x) to zero and solve for x:
Setting ∂f/∂x = 7x^6 equal to zero, we get:
7x^6 = 0
Dividing both sides by 7, we have:
x^6 = 0
Since any number raised to the power of 6 will never equal zero, there are no real x-values that satisfy this equation. Therefore, there is no specific value for x in f(x,y) = x^7 - 8y - 3.
Step 3: Calculate the partial derivative with respect to y (∂f/∂y):
To take the partial derivative with respect to y, treat x as a constant and differentiate -8y. The derivative of -8y is simply -8.
Step 4: Set the partial derivative with respect to y (∂f/∂y) to zero and solve for y:
Setting ∂f/∂y = -8 equal to zero, we have:
-8 = 0
Since -8 is never equal to zero, there is no specific value for y in f(x,y) = x^7 - 8y - 3.
Therefore, in the equation f(x,y) = x^7 - 8y - 3, there are no specific values for x and y that satisfy the equation.
Question 2: Find x and y in the equation f(x,y) = 7 / (5x + 4y).
To find x and y in this equation, we will again set up partial derivatives. The partial derivative with respect to x (∂f/∂x) and the partial derivative with respect to y (∂f/∂y) will help us find x and y, respectively.
Step 1: Calculate the partial derivative with respect to x (∂f/∂x):
To take the partial derivative with respect to x, treat y as a constant and differentiate 7 / (5x + 4y). The derivative of 7 / (5x + 4y) with respect to x is (-35) / (5x + 4y)^2.
Step 2: Set the partial derivative with respect to x (∂f/∂x) to zero and solve for x:
Setting ∂f/∂x = (-35) / (5x + 4y)^2 equal to zero, we get:
(-35) / (5x + 4y)^2 = 0
This equation is satisfied when the numerator (-35) equals zero, since any non-zero number divided by zero is undefined. However, -35 is not equal to zero. Therefore, there is no specific value for x in f(x,y) = 7 / (5x + 4y) that makes the partial derivative with respect to x equal to zero.
Step 3: Calculate the partial derivative with respect to y (∂f/∂y):
To take the partial derivative with respect to y, treat x as a constant and differentiate 7 / (5x + 4y). The derivative of 7 / (5x + 4y) with respect to y is (-28) / (5x + 4y)^2.
Step 4: Set the partial derivative with respect to y (∂f/∂y) to zero and solve for y:
Setting ∂f/∂y = (-28) / (5x + 4y)^2 equal to zero, we get:
(-28) / (5x + 4y)^2 = 0
This equation is satisfied when the numerator (-28) equals zero, since any non-zero number divided by zero is undefined. However, -28 is not equal to zero. Therefore, there is no specific value for y in f(x,y) = 7 / (5x + 4y) that makes the partial derivative with respect to y equal to zero.
Therefore, in the equation f(x,y) = 7 / (5x + 4y), there are no specific values for x and y that satisfy the equation.