Chapter 14, Problem 8T

### Mathematical Applications for the ...

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

Chapter
Section

### Mathematical Applications for the ...

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

# Suppose a store sells two brands of disposable razors and the profit for these is a function of their two selling prices. The type 1 razor sells for $x, the type 2 sells for$y, and profit is given by P = 915 x − 30 x 2 − 45 x y + 975 y − 30 y 2 − 3500 Find the selling prices that maximize profit. Find the maximum profit.

To determine

To calculate: The selling prices that maximize profit and the maximum profit where a store sells two brands of disposable razors and the profit for these is a function of their two selling prices. The type 1 razor sells for $x, the type 2 sells for$y and the profit is given by P=915x30x245xy+975y30y23500.

Explanation

Given Information:

A store sells two brands of disposable razors and the profit for these is a function of their two selling prices. The type 1 razor sells for $x, the type 2 sells for$y and the profit is given by P=915xâˆ’30x2âˆ’45xy+975yâˆ’30y2âˆ’3500.

Formula used:

To calculate relative maxima and minima of the z=f(x,y),

(1) Find the partial derivatives âˆ‚zâˆ‚x and âˆ‚zâˆ‚y.

(2) Find the critical points, that is, the point(s) that satisfy âˆ‚zâˆ‚x=0 and âˆ‚zâˆ‚y=0.

(3) Then find all the second partial derivatives and evaluate the value of D at each critical point, where D=(zxx)(zyy)âˆ’(zxy)2=âˆ‚2zâˆ‚x2â‹…âˆ‚2zâˆ‚y2âˆ’(âˆ‚2zâˆ‚xâˆ‚y)2.

(a) If D>0, then relative minimum occurs if zxx>0 and relative maximum occurs if zxx<0.

(b) If D<0, then neither a relative maximum nor a relative minimum occurs.

For a function f(x,y), the partial derivative of f with respect to x is calculated by taking the derivative of f(x,y) with respect to x and keeping the other variable y constant and the partial derivative of f with respect to y is calculated by taking the derivative of f(x,y) with respect to y and keeping the other variable x constant. The partial derivative of f with respect to x is denoted by fx and with respect to y is denoted by fy.

For a function z(x,y), the second partial derivative,

1. When both derivatives are taken with respect to x is zxx=âˆ‚2zâˆ‚x2=âˆ‚âˆ‚x(âˆ‚zâˆ‚x).

2. When both derivatives are taken with respect to y is zyy=âˆ‚2zâˆ‚y2=âˆ‚âˆ‚y(âˆ‚zâˆ‚y).

3. When first derivative is taken with respect to x and second derivative is taken with respect to y is zxy=âˆ‚2zâˆ‚yâˆ‚x=âˆ‚âˆ‚y(âˆ‚zâˆ‚x).

4. When first derivative is taken with respect to y and second derivative is taken with respect to x is zyx=âˆ‚2zâˆ‚xâˆ‚y=âˆ‚âˆ‚x(âˆ‚zâˆ‚y).

Power of x rule for a real number n is such that, if f(x)=xn then fâ€²(x)=nxnâˆ’1.

Chain rule for function f(x)=u(v(x)) is fâ€²(x)=uâ€²(v(x))â‹…vâ€²(x).

Constant function rule for a constant c is such that, if f(x)=c then fâ€²(x)=0.

Coefficient rule for a constant c is such that, if f(x)=câ‹…u(x), where u(x) is a differentiable function of x, then fâ€²(x)=câ‹…uâ€²(x).

Calculation:

Consider the problem, the type 1 razor sells for $x, the type 2 sells for$y, and the profit is given by P=915xâˆ’30x2âˆ’45xy+975yâˆ’30y2âˆ’3500.

The provided function is P(x,y)=915xâˆ’30x2âˆ’45xy+975yâˆ’30y2âˆ’3500.

Use the power of x rule for derivatives, the constant function rule, the chain rule, and the coefficient rule,

Thus,

âˆ‚Pâˆ‚x=0915âˆ’30(2x)âˆ’45y=045y=915âˆ’60xy=183âˆ’12x9

And,

âˆ‚Pâˆ‚y=0975âˆ’30(2y)âˆ’45x=060y=975âˆ’45xy=195âˆ’9x12

Now, calculate the values of x and y

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