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Chapter 3: Problem 61
Prove that \(|\cos a-\cos b| \leq|a-b|\) for all \(a\) and \(b\).
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Assume that \(f\) is differentiable for all \(x\). The signs of \(f^{\prime}\) are as follows. \(f^{\prime}(x)>0\) on \((-\infty,-4)\) \(f^{\prime}(x)<0\) on (-4,6) \(f^{\prime}(x)>0\) on \((6, \infty)\) Supply the appropriate inequality for the indicated value of \(c\). $$ g(x)=-f(x) \quad g^{\prime}(-6) \quad 0 $$
In Exercises \(57-74\), sketch the graph of the equation. Look for extrema, intercepts, symmetry, and asymptotes as necessary. Use a graphing utility to verify your result. $$ y=\frac{x^{3}}{\sqrt{x^{2}-4}} $$
Numerical, Graphical, and Analytic Analysis Consider the functions \(f(x)=x\)
and \(g(x)=\tan x\) on the interval \((0, \pi / 2)\)
(a) Complete the table and make a conjecture about which is the greater
function on the interval \((0, \pi / 2)\).
$$
\begin{array}{|l|l|l|l|l|l|l|}
\hline x & 0.25 & 0.5 & 0.75 & 1 & 1.25 & 1.5 \\
\hline f(x) & & & & & & \\
\hline g(x) & & & & & & \\
\hline
\end{array}
$$
(b) Use a graphing utility to graph the functions and use the graphs to make a
conjecture about which is the greater function on the interval \((0, \pi / 2)\).
(c) Prove that \(f(x)
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the graph of a polynomial function has three \(x\) -intercepts, then it must have at least two points at which its tangent line is horizontal.
In Exercises 71 and \(72,\) let \(f\) and \(g\) represent differentiable functions such that \(f^{\prime \prime} \neq 0\) and \(g^{\prime \prime} \neq 0\). Show that if \(f\) and \(g\) are concave upward on the interval \((a, b)\), then \(f+g\) is also concave upward on \((a, b)\).
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