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Chapter 3: Q. 5 (page 291)

Consider the graph of the function f shown next. Define A(x) to be the area of the region between the graph of f and the x-axis from 0 to x. We will count areas of regions above
the x-axis positively and areas of regions below the x-axis negatively.

Show that your answers for the local and global extrema of A(x) are reasonable by using optimization techniques on the area function A(x)=3x4-32x3+90x2?

Short Answer

Expert verified

The global maxima of A(x)is A(3)=189and the global minima is A(0)=0

Step by step solution

01

Step 1. Given Information.

The graph.

The area function:

A(x)=3x4-32x3+90x2

02

Step 2. Find the critical points.

Differentiate with respect to x,

A(x)=3x4-32x3+90x2A'(x)=12x3-96x2+180x(x2-8x+15)12x=0x(x-5)(x-3)=0x=0,3,5

Then x=0,3is the local minimum and x=5is the local maximum.

03

Step 3. Values of A(x).

The value of A(x) are:

A(0)=0A(1)=61A(2)=152A(3)=189A(4)=160A(5)=125

The global maxima of A(x)is A(3)=189and the global minima is A(0)=0

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