Homework # 14 - Math 149
The problems on this homework must be done alone. The honor code is in effect.
First name: Last name:
1. Consider the function \(f(x) = x^2\), on the interval \( (-1, 2)\).
a. This function has no maximum and no minimum on this interval.
b. This function has a maximum but no minimum on this interval.
c. This function has a minimum but no maximum on this interval.
d. This function has a minimum and a maximum on this interval.
2. Consider the function \(f(x) = x^2\), on the interval \( [-2, 1)\).
3. Find all the critical values of the function \(g(x) = x^3-3x^2-9x\).
a. \(x = 0\) only.
b. \(x = -1\) only.
c. \(x = 0\) and \(x = 3\).
d. \(x = -1\) and \(x = 3\).
e. None of the above.
4. Can you use the Intermediate Value Theorem to prove that the function \(f(x) = \frac{1}{x-2}\) has a zero (some x value where the function equals zero) somewhere between x = 1 and x = 3?
a. Yes.
b. No, because the function \(f\) is not continuous on the interval \( [1, 3] \).
c. No, because 0 isn't between the values \( f(1)\) and \( f(3)\).
5. Can you use the Intermediate Value Theorem to prove that the function \(f(x) = \frac{1}{x-2}\) has a zero (some x value where the function equals zero) somewhere between x = -1 and x = 1?
b. No, because the function \(f\) is not continuous on the interval \( [-1, 1] \).
c. No, because 0 isn't between the values \( f(-1)\) and \( f(1)\).
6. Can you use the Intermediate Value Theorem to prove that the function \(f(x) = \frac{x^3+x+1}{x-2}\) has a zero (some x value where the function equals zero) somewhere between x = -1 and x = 1?