Chapter 5

\(f(x)= \begin{cases}(x+9)^{2}-8 & \text { if } x<-7 \\ -\sqrt{25-(x+4)^{2}} & \text { if }-7 \leq x \leq 0 \\ (x-2)^{2}-7 & \text { if } 0

\(\lim _{x \rightarrow c^{-}} f^{\prime}(x)=+\infty ; \lim _{x \rightarrow c^{+}} f^{\prime}(x)=0 ; f^{\prime \prime}(x)>0\) if \(xc\)

\(f(x)=2+(x-3)^{1 / 3}\)

The demand equation for a certain commodity produced by a monopolist is \(p=a-b x\), and the total cost, \(C(x)\) dollars, of producing \(x\) units is determined by \(C(x)=c+d x\), where \(a, b, c\), and \(d\) are positive constants. If the government levies a tax on the monopolist of \(t\) dollars per unit produced, show that in order for the monopolist to maximize his profits he should pass on to the consumer only one-half of the tax; that is, he should increase his unit price by \(\frac{1}{2} t\) dollars.

Prove by the method of this section that the shortest distance from the point \(P_{1}\left(x_{1}, y_{1}\right)\) to the line \(l\), having the equation \(A x+B y+C=0\), is \(\left|A x_{1}+B y_{1}+C\right| / \sqrt{A^{2}+\bar{B}^{2}} .\) (HINT: If \(s\) is the number of units from \(P_{1}\) to a point \(P(x, y)\) on \(l\), then \(s\) will be an absolute minimum when \(s^{2}\) is an absolute minimum.)

\(\lim _{x \rightarrow c-} f^{\prime}(x)=+\infty ; \lim _{x \rightarrow c^{+}} f^{\prime}(x)=-\infty ; f^{\prime \prime}(x)>0\) if \(x0\) if \(x>c\)

\(f(x)=2+(x-3)^{4 / 3}\)

Draw a sketch of the graph of a function \(f\) for which \(f(x), f^{\prime}(x)\), and \(f^{\prime \prime}(x)\) exist and are positive for all \(x\).

\(f(x)=2+(x-3)^{5 / 3}\)

\(f(x)=2+(x-3)^{2 / 3}\)

\(f(x)=3+(x+1)^{6 / 5}\)

Suppose that \(f\) is a function for which \(f^{\prime \prime}(x)\) exists for all values of \(x\) in some open interval \(I\) and that at a number \(c\) ir I, \(f^{\prime \prime}(c)=0\) and \(f^{\prime \prime \prime}(c)\) exists and is not zero. Prove that the point \((c, f(c))\) is a point of inflection of the graph of \(f\) (HINT: The proof is similar to the proof of the second-derivative test (Theorem 5.2.1).)

\(f(x)=3+(x+1)^{7 / 5}\)

\(f(x)=(x-a)^{2 / 5}+1\)

Find \(a\) and \(b\) so that the function defined by \(f(x)=x^{3}+a x^{2}+b\) will have a relative extremum at \((2,3)\).

\(f(x)=\frac{(x+1)^{2}}{x^{2}+1}\)

Find \(a, b\), and \(c\) so that the function defined by \(f(x)=a x^{2}+b x+c\) will have a relative maximum value of 7 at 1 and the graph of \(y=f(x)\) will go through the point \((2,-2)\).

Find \(a, b, c\), and \(d\) so that the function defined by \(f(x)=a x^{3}+b x^{2}+c x+d\) will have relative extrema at \((1,2)\) and \((2,3)\).

Given \(f(x)=x^{p}(1-x)^{q}\), where \(p\) and \(q\) are positive integers greater than 1, prove each of the following: (a) if \(p\) is even, \(f\) has a relative minimum value at 0 ; (b) if \(q\) is even, \(f\) has a relative minimum value at 1 ; (c) \(f\) has a relative maximum value at \(p /(p+q)\) whether \(p\) and \(q\) are odd or even.

Prove that if \(f\) is increasing on \([a, b]\) and if \(g\) is increasing on \([f(a), f(b)]\), then if \(g \circ f\) exists on \([a, b], g \circ f\) is increasing on \([a, b]\)

The function \(f\) is increasing on the interval \(I\). Prove: (a) if \(g(x)=-f(x)\), then \(g\) is decreasing on \(I ;(b)\) if \(h(x)=1 / f(x)\), and \(f(x)>0\) on \(I\), then \(h\) is decreasing on \(I\).

The function \(f\) is differentiable at each number in the closed interval \([a, b] .\) Prove that if \(f^{\prime}(a) \cdot f^{\prime}(b)<0\), there is a number \(c\) in the open interval \((a, b)\) such that \(f^{\prime}(c)=0\).