Question: x 0 0 < x < 1 1 1 < x < 2 2 2 < x < 3 3 3 < x < 4 4
f(x) 0 negative and decreasing -1 negative and decreasing -3 negative and increasing 0 positive and increasing 2
The polynomial function f is an odd function with domain $$-4 \leq x \leq 4$$. The table gives information
about values of $$f(x)$$ and the behavior of the function.
What is the absolute maximum value of $$f$$ on its domain?
A 4
B 3
C 2
D 0
1. **State the problem:** We are given a polynomial function $$f$$ defined on the domain $$-4 \leq x \leq 4$$, which is an odd function. The table provides values and behavior of $$f(x)$$ on intervals between integers.
2. **Recall properties of odd functions:** For an odd function, $$f(-x) = -f(x)$$ for all $$x$$ in the domain. This means the function is symmetric about the origin.
3. **Analyze the given data:**
- At $$x=0$$, $$f(0) = 0$$.
- For $$0 < x < 1$$, $$f(x)$$ is negative and decreasing.
- At $$x=1$$, $$f(1) = -1$$.
- For $$1 < x < 2$$, $$f(x)$$ is negative and decreasing.
- At $$x=2$$, $$f(2) = -3$$.
- For $$2 < x < 3$$, $$f(x)$$ is negative and increasing.
- At $$x=3$$, $$f(3) = 0$$.
- For $$3 < x < 4$$, $$f(x)$$ is positive and increasing.
- At $$x=4$$, $$f(4) = 2$$.
4. **Use odd function property to find values for negative $$x$$:**
- $$f(-1) = -f(1) = -(-1) = 1$$
- $$f(-2) = -f(2) = -(-3) = 3$$
- $$f(-3) = -f(3) = -0 = 0$$
- $$f(-4) = -f(4) = -2$$
5. **Determine absolute maximum:**
- From the values, the maximum values at negative $$x$$ are $$f(-1) = 1$$ and $$f(-2) = 3$$.
- At positive $$x$$, the maximum value is $$f(4) = 2$$.
- At zero, $$f(0) = 0$$.
6. **Conclusion:** The absolute maximum value of $$f$$ on $$[-4,4]$$ is $$3$$ at $$x = -2$$.
**Final answer:** B 3