91Ó°ÊÓ

Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. $$f(x)=\tan ^{-1} x$$

Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. $$f(x)=2 x^{4}+8 x^{3}+12 x^{2}-x-2$$

Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. $$f(x)=x^{3}-3 x^{2}$$

Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. $$f(x)=2 x^{3}-3 x^{2}+12$$

Is it possible? Determine whether the following properties can be satisfied by a function that is continuous on \((-\infty, \infty)\). If such a function is possible, provide an example or a sketch of the function. If such a function is not possible, explain why. a. A function \(f\) is concave down and positive everywhere. b. A function \(f\) is increasing and concave down everywhere. c. A function \(f\) has exactly two local extrema and three inflection points. d. A function \(f\) has exactly four zeros and two local extrema.

Use a graphing utility together with analytical methods to create a complete graph of the following functions. Be sure to find and label the intercepts, local extrema, inflection points, asymptotes, intervals where the function is increasing/decreasing, and intervals of concavity.

Concavity of parabolas Consider the general parabola described by the function \(f(x)=a x^{2}+b x+c .\) For what values of \(a, b\) and \(c\) is \(f\) concave up? For what values of \(a, b,\) and \(c\) is \(f\) concave down?

General quartic Show that the general quartic (fourth-degree) polynomial \(f(x)=x^{4}+a x^{3}+b x^{2}+c x+d\) has either zero or two inflection points, and that the latter case occurs provided that \(b<3 a^{2} / 8\)

The complexity of a computer algorithm is the number of operations or steps the algorithm needs to complete its task assuming there are \(n\) pieces of input (for example, the number of steps needed to put \(n\) numbers in ascending order). Four algorithms for doing the same task have complexities of A: \(n^{3 / 2}, B: n \log _{2} n, C: n\left(\log _{2} n\right)^{2},\) and \(D: \sqrt{n} \log _{2} n .\) Rank the algorithms in order of increasing efficiency for large values of \(n\) Graph the complexities as they vary with \(n\) and comment on your observations.

Show that any exponential function \(b^{x}\), for \(b>1,\) grows faster than \(x^{p},\) for \(p>0\)