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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 \( [-2, 1)\).

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.

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?

a.  Yes.

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?

a.  Yes.

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 \( f(1)\).

7. Can you apply Rolle's Theorem to the function \(f(x) = \mid x \mid\) on the interval \( [-1, 1]\) to find an \(x\)-value \(c\) where \(f'(c) = 0\)?

a.  No, because \(f\) is not continuous on \([-1, 1]\).

b.  No, because \(f\) is not differentiable on \((-1, 1)\).

c.  No, because \(f(1) \neq f(-1)\)

d.  Yes.

8.  Which of the following can't happen for a continuous function, \(f\),on a open interval \( (a,b)\)?

a.  The function has no critical values on this interval.

b.  The function has 10 different relative maxima on the interval.

c.  The function has three different relative maxima but no relative minima on the interval.

d.  Every point on the interval is both a relative maximum and a relative minimum.

e.  The function has a critical point which is neither a relative maximum nor a relative minimum.

9.  The minimum of \(f(x) = x^3 - 6x + 4\) on the interval \([0, 2]\) occurs at

a.  \(x = 0\).

b.  \(x = 1\).

c.  \(x = \sqrt{2}\).

d.  \(x = \sqrt{3}\).

e.  \(x = 2\).