91Ó°ÊÓ

Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the \(x\) -value at which each extremum occurs. When no interval is specified, use the real numbers, \((-\infty, \infty)\). $$f(x)=x^{2}+\frac{432}{x} ; \quad(0, \infty)$$

Look up "differential" in a book or Web site devoted to math history. In a short paragraph, describe your findings.

Assume that \(f\) is differentiable over \((-\infty, \infty) .\) Classify each of the following statements as either true or false. If a statement is false, explain why. If \(f\) has exactly two critical values at \(x=a\) and \(x=b\), where \(a

Find each limit, if it exists. $$ \lim _{x \rightarrow-2} \frac{x^{3}+8}{x^{2}-4} $$

Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the \(x\) -value at which each extremum occurs. When no interval is specified, use the real numbers, \((-\infty, \infty)\). $$f(x)=x^{2}+\frac{250}{x} ; \quad(0, \infty)$$

Find each limit, if it exists. $$ \lim _{x \rightarrow-\infty} \frac{-6 x^{3}+7 x}{2 x^{2}-3 x-10} $$

For Exercises 69-84, draw a graph to match the description given. Answers will vary. \(f(x)\) is increasing over \((-\infty, 2)\) and decreasing over \((2, \infty)\)

Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the \(x\) -value at which each extremum occurs. When no interval is specified, use the real numbers, \((-\infty, \infty)\). $$f(x)=2 x^{4}+x ; \quad[-1,1]$$

Assume that \(f\) is differentiable over \((-\infty, \infty) .\) Classify each of the following statements as either true or false. If a statement is false, explain why. A function \(f\) can have no extrema but still have at least one point of inflection.

Find each limit, if it exists. $$ \lim _{x \rightarrow 1} \frac{x^{3}-1}{x^{2}-1} $$