Let f be a polynomia I function and let f' its derivative be defined as f'(x)=-x(x+2)(x-2) At how many points does the graph of f have a relative maximum?

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Let f be a polynomia I function and let f' its derivative be defined as f'(x)=-x(x+2)(x-2) At how many points does the graph of f have a relative maximum?

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Answer 1

To determine the number of relative maxima of the polynomial function \( f \), we need to analyze its derivative \( f'(x) \).

Given:
\[ f'(x) = -x(x+2)(x-2) \]

First, let's find the critical points by setting \( f'(x) = 0 \):
\[ -x(x+2)(x-2) = 0 \]

This equation is satisfied when any of the factors are zero:
\[ x = 0, \quad x + 2 = 0 \implies x = -2, \quad x - 2 = 0 \implies x = 2 \]

So, the critical points are \( x = 0 \), \( x = -2 \), and \( x = 2 \).

Next, we need to determine the nature of these critical points (whether they are relative maxima or minima). We can do this by examining the sign changes of \( f'(x) \) around these points.

1. For \( x = -2 \ - Choose a test point \( x = -3 \):
\[ f'(-3) = -(-3)((-3)+2)((-3)-2) = -(-3)(-1)(-5) = -15 \]
(Negative)

- Choose a test point \( x = -1 \):
\[ f'(-1) = -(-1)((-1)+2)((-1)-2) = -(-1)(1)(-3) = 3 \]
(Positive)

Since \( f'(x) \) changes from negative to positive at \( x = -2 \), this indicates a relative minimum.

2. For \( x = 0 \):
- Choose a test point \( x = -1 \):
\[ f'(-1) = -(-1)((-1)+2)((-1)-2) = -(-1)(1)(-3) = 3 \]
(Positive)

- Choose a test point \( x = 1 \):
\[ f'(1) = -(1)((1)+2)((1)-2) = -(1)(3)(-1) = 3 \]
(Positive)

Since \( f'(x) \) does not change sign at \( x = 0 \), this indicates neither a relative maximum nor minimum.

3. For \( x = 2 \):
- Choose a test point \( x = 3 \):
\[ f'(3) = -(3)((3)+2)((3)-2) = -(3)(5)(1) = -15 \]
(Negative)

- Choose a test point \( x = 4 \):
\[ f'(4) = -(4)((4)+2)((4)-2) = -(4)(6)(2) = -48 \]
(Negative)

Since \( f'(x) \) remains negative around \( x = 2 \), this indicates a relative minimum.

Therefore, the graph of \( f \) has no relative maxima.

Pertanyaan

Let f be a polynomia I function and let f' its derivative be defined as f'(x)=-x(x+2)(x-2) At how many points does the graph of f have a relative maximum?

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