Chapter 4

Show that the equation \(x^{4}+4 x+c=0\) has at most two real roots.

\(15-22=\) Sketch the graph of \(f\) by hand and use your sketch to find the absolute and local maximum and minimum values of \(f .\) (Use the graphs and transformations of Sections \(1.2 . )\) $$f(x)=1-\sqrt{x}$$

Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of radius \(r .\)

Sketch the graph of a function that satisfies all of the given conditions. \(f^{\prime}(x)>0\) if \(|x|<2, \quad f^{\prime}(x)<0\) if \(|x|>2\) \(f^{\prime}(-2)=0, \quad \lim _{x \rightarrow 2}\left|f^{\prime}(x)\right|=\infty, \quad f^{\prime \prime}(x)>0\) if \(x \neq 2\)

(a) Show that a polynomial of degree 3 has at most three real roots. (b) Show that a polynomial of degree \(n\) has at most \(n\) real roots.

Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 3 \(\mathrm{cm}\) and 4 \(\mathrm{cm}\) if two sides of the rectangle lie along the legs.

Sketch the graph of a function that satisfies all of the given conditions. \(f^{\prime}(x)>0\) if \(|x|<2, \quad f^{\prime}(x)<0\) if \(|x|>2,\) \(f^{\prime}(2)=0, \quad \lim _{x \rightarrow \infty} f(x)=1, \quad f(-x)=-f(x),\) \(f^{\prime \prime}(x) <0\) if \(0 < x < 3, \quad f^{\prime \prime}(x) > 0\) if \(x > 3\)

\(15-22=\) Sketch the graph of \(f\) by hand and use your sketch to find the absolute and local maximum and minimum values of \(f .\) (Use the graphs and transformations of Sections \(1.2 . )\) $$f(x)=\left\\{\begin{array}{ll}{4-x^{2}} & {\text { if }-2 \leqslant x<0} \\\ {2 x-1} & {\text { if } 0 \leqslant x \leqslant 2}\end{array}\right.$$

Use Newton's method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. \(e^{\arctan x}=\sqrt{x^{3}+1}\)

(a) Suppose that \(f\) is differentiable on \(\mathbb{R}\) and has two roots. Show that \(f^{\prime}\) has at least one root. (b) Suppose \(f\) is twice differentiable on \(\mathbb{R}\) and has three roots. Show that \(f^{\prime \prime}\) has at least one real root. (c) Can you generalize parts (a) and (b)?

Find \(f\) $$f^{\prime}(x)=1+3 \sqrt{x}, \quad f(4)=25$$

\(23-36=\) Find the critical numbers of the function. $$f(x)=4+\frac{1}{3} x-\frac{1}{2} x^{2}$$

A right circular cylinder is inscribed in a sphere of radius \(r .\) Find the largest possible volume of such a cylinder.

(a) Apply Newton's method to the equation \(x^{2}-a=0\) to derive the following square-root algorithm (used by the ancient Babylonians to compute \(\sqrt{a}\) ): $$x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{a}{x_{n}}\right)$$ (b) Use part (a) to compute \(\sqrt{1000}\) correct to six decimal places.

If \(f(1)=10\) and \(f^{\prime}(x) \geqslant 2\) for \(1 \leqslant x \leqslant 4,\) how small can \(f(4)\) possibly be?

Find \(f\) $$f^{\prime}(x)=5 x^{4}-3 x^{2}+4, \quad f(-1)=2$$

\(23-36=\) Find the critical numbers of the function. $$f(x)=x^{3}+6 x^{2}-15 x$$

Find the area of the largest rectangle that can be inscribed in the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\)

(a) Apply Newton's method to the equation \(1 / x-a=0\) to derive the following reciprocal algorithm: $$x_{n+1}=2 x_{n}-a x_{n}^{2}$$ (This algorithm enables a computer to find reciprocals without actually dividing.) (b) Use part (a) to compute 1\(/ 1.6984\) correct to six decimal places.

Suppose that 3\(\leqslant f^{\prime}(x) \leqslant 5\) for all values of \(x .\) Show that \(18 \leqslant f(8)-f(2) \leqslant 30\)

Find \(f\) $$f^{\prime}(t)=4 /\left(1+t^{2}\right), \quad f(1)=0$$

\(23-36=\) Find the critical numbers of the function. $$f(x)=2 x^{3}-3 x^{2}-36 x$$

Explain why Newton's method doesn't work for finding the root of the equation \(x^{3}-3 x+6=0\) if the initial approximation is chosen to be \(x_{1}=1\)

A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 30 \(\mathrm{ft}\) , find the dimensions of the window so that the greatest possible amount of light is admitted.

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the inforvation from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one. \(f(x)=x^{3}-12 x+2\)

Does there exist a function \(f\) such that \(f(0)=-1\) \(f(2)=4,\) and \(f^{\prime}(x) \leqslant 2\) for all \(x ?\)

Find \(f\) $$f^{\prime}(t)=2 \cos t+\sec ^{2} t, \quad-\pi / 2

\(23-36=\) Find the critical numbers of the function. $$f(x)=2 x^{3}+x^{2}+2 x$$

A right circular cylinder is inscribed in a cone with height \(h\) and base radius \(r .\) Find the largest possible volume of such a cylinder.

(a) Use Newton's method with \(x_{1}=1\) to find the root of the equation \(x^{3}-x=1\) correct to six decimal places. (b) Solve the equation in part (a) using \(x_{1}=0.6\) as the initial approximation. (c) Solve the equation in part (a) using \(x_{1}=0.57 .\) (You definitely need a programmable calculator for this part.)

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the inforvation from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one. \(f(x)=36 x+3 x^{2}-2 x^{3}\)

Suppose that \(f\) and \(g\) are continuous on \([a, b]\) and differentiable on \((a, b) .\) Suppose also that \(f(a)=g(a)\) and \(f^{\prime}(x)

Find \(f\) $$f^{\prime}(t)=2 \cos t+\sec ^{2} t, \quad-\pi / 2

\(23-36=\) Find the critical numbers of the function. $$g(t)=t^{4}+t^{3}+t^{2}+1$$

A piece of wire 10 \(\mathrm{m}\) long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is (a) a maximum? (b) A minimum?

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the inforvation from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one. \(f(x)=2+2 x^{2}-x^{4}\)

Show that \(\sqrt{1+x}<1+\frac{1}{2} x\) if \(x>0\)

Find \(f\) $$f^{\prime}(x)=4 / \sqrt{1-x^{2}}, \quad f\left(\frac{1}{2}\right)=1$$

Use Newton's method to find the absolute maximum value of the function \(f(x)=x \cos x, 0 \leqslant x \leqslant \pi,\) correct to six decimal places.

A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the inforvation from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one. \(g(x)=200+8 x^{3}+x^{4}\)

\(23-36=\) Find the critical numbers of the function. $$g(t)=|3 t-4|$$

Suppose \(f\) is an odd function and is differentiable everywhere. Prove that for every positive number \(b,\) there exists a number \(c\) in \((-b, b)\) such that \(f^{\prime}(c)=f(b) / b\)

Find \(f\) $$f^{\prime \prime}(x)=-2+12 x-12 x^{2}, \quad f(0)=4, \quad f^{\prime}(0)=12$$

\(23-36=\) Find the critical numbers of the function. $$g(y)=\frac{y-1}{y^{2}-y+1}$$

Use Newton's method to find the coordinates of the inflection point of the curve \(y=x^{2} \sin x, 0 \leqslant x \leqslant \pi,\) correct to six decimal places.

Use the guidelines of this section to sketch the curve. $$y=x \tan x, \quad-\pi / 2 < x < \pi / 2$$

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the inforvation from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one. \(h(x)=(x+1)^{5}-5 x-2\)

Use the Mean Value Theorem to prove the inequality $$|\sin a-\sin b| \leqslant|a-b| \quad\( for all \)a\( and \)b$$

Find \(f\) $$f^{\prime \prime}(x)=8 x^{3}+5, \quad f(1)=0, \quad f^{\prime}(1)=8$$