Question

A) Graph the function and label all zeros. Be sure to label the axis and what you are incrementing by.

Answer 1

9514 1404 393

Answer:

• graph is shown below
• absolute max and min do not exist
• local max: 0 at x=0
• local min: -500/27 ≈ -18.519 at x=10/3

Step-by-step explanation:

The function is odd degree so has no absolute maximum or minimum. It factors as ...

  g(x) = x^2(x -5)

so has zeros at x=0 (multiplicity 2, meaning this is a local maximum*) and x=5.

Differentiating, we find the derivative of g(x) is zero at x = 0 and x = 10/3.

  g'(x) = 3x^2 -10x = x(3x -10)   ⇒   x=0 and x=10/3 are critical points

The value of g(10/3) is a local minimum. That value is ...

  g(10/3) = (10/3)^2((10-15)/3) = -500/27 ≈ -18.519

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The local maximum is (0, 0); the local minimum is (10/3, -500/27). The graph is shown below.

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* When a root has even multiplicity, the graph does not cross the x-axis. That means the root corresponds to a local extremum. Since this is the left-most root of an odd-degree function with a positive leading coefficient, it is a local maximum. (The function is increasing left of the left-most turning point.)