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Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

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BuyFindarrow_forward

Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

Suppose we want to find the minimum value of C   = 5 x   +   2 y and we know that x and y must be positive and that x y =   1000 .

Find the minimum value of C.

To determine

To calculate: The minimum value of the equation C=5x+2y when x,y are positive and xy=1000.

Explanation

Given Information:

The equation is C=5x+2y when x,y are positive and xy=1000.

Formula used:

To find the minimum value, first calculate the relevant stationary value of an equation. Differentiate the function with respect to the independent variable and equate it to 0. The relevant value is the stationary value that satisfies the provided conditions.

Next, calculate the second derivative of the function and substitute the relevant stationary value. If it is greater than 0, find the minimum value of the equation by substituting the relevant stationary value. Otherwise, it is not the minimum value.

The power rule is used for a function in which the expression can be written as every term raised to a power (be it fractional, positive or negative). For the function f(x)=xn, the derivative is

ddx[xn]=nxn1

Calculation:

Consider the provided equation:

C=5x+2y

Where x,y are positive and xy=1000.

Apply the equation C=5x+2y and substitute y=1000x to get

C=5x+2(1000x)C=5x+2000x

To find the critical values for C=5x+2000x, differentiate the equation with respect to the independent variable x to get

C'(x)=52000x2

Put C'(x)=0 to get x,

52000

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