Identifying Extrema
In Exercises 19–40:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local extreme values, if any, saying where they occur.
f(x) = (x² − 3) / (x − 2), x ≠ 2
Checking the Mean Value Theorem
Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.
f(x) = √(x(1 − x)), [0, 1]
In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
53. y = x * √(8 - x²)
Business and Economics
60. Production level Prove that the production level (if any) at which average cost is smallest is a level at which the average cost equals marginal cost.
Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
g(t) = √t + √(1 + t) − 4, (0, ∞)
93. The accompanying figure shows a portion of the graph of a twice-differentiable function y=f(x). At each of the five labeled points, classify y' and \(\y\)'' as positive, negative, or zero.
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