   Chapter 10, Problem 51RE ### Mathematical Applications for the ...

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

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

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

# Drug sensitivity The reaction R to an injection of a drug is related to the dose x, in milligrams, according to R ( x ) = x 2 ( 500 − x 3 ) The sensitivity to the drug is defined by dR/dx. Find the dose that maximizes sensitivity.

To determine

To calculate: The dose that maximizes sensitivity, if the reaction R to an injection of a drug is related to the dose x, in milligrams, according to R(x)=x2(500x3) and the sensitivity to the drug is defined by dRdx.

Explanation

Given Information:

The provided equation is R(x)=x2(500x3), where the reaction R to an injection of a drug is related to the dose x.

Formula Used:

The critical values are the only values at which the graph can have turning points, the derivative cannot change sign anywhere except at the critical value.

This, in an interval between two critical values, the sign of the derivative at any value in the interval will be the sign of the derivative at all values in the interval.

As per the First Derivative Test,

The first derivative of the function is evaluated. The first derivative is made equal to zero in order to get the critical points.

The values of the critical values are kept inside the original function which gives the critical points. The intervals of the values of x are then evaluated for the relative maximum and minimum.

Calculation:

Consider The provided equation is R(x)=x2(500x3),

In order to minimize the sensitivity, the following steps have to followed:

As per the given information:

Simplify the equation as follows:

R(x)=x2(500x3)=500x2x33

Take out the first derivative of the equation by the power rule,

R=ddx(500x2x33)=500ddx(x2)ddx(x33)=1000xx2

The second derivative is as follows:

R=ddx

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