Chapter 10

Use the information summarized in the table to sketch the graph of \(f\). \(f(x)=\frac{1}{9}\left(x^{4}-4 x^{3}\right)\) Domain: \((-\infty, \infty)\) Intercepts: \(x\) -intercepts: 0,\(4 ; y\) -intercept: 0 Asymptotes: None Intervals where \(f\) is \(\nearrow\) and \(\searrow: \nearrow\) on \((3, \infty)\); \(\searrow\) on \((-\infty, 0) \cup(0,3)\) Relative extrema: Rel. min. at \((3,-3)\) Concavity: Downward on \((0,2) ;\) upward on \((-\infty, 0) \cup(2, \infty)\) Points of inflection: \((0,0)\) and \(\left(2,-\frac{16}{9}\right)\)

Determine where the function is concave upward and where it is concave downward. $$ g(x)=\sqrt{x-2} $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ g(x)=x \sqrt{x+1} $$

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=e^{-x^{2}} \text { on }[-1,1] $$

Use the information summarized in the table to sketch the graph of \(f\). \(f(x)=\frac{4 x-4}{x^{2}}\) Domain: \((-\infty, 0) \cup(0, \infty)\) Intercept: \(x\) -intercept: 1 Asymptotes: \(x\) -axis and \(y\) -axis Intervals where \(f\) is \(\nearrow\) and \(\searrow: \nearrow\) on \((0,2)\); \(\searrow\) on \((-\infty, 0) \cup(2, \infty)\) Relative extrema: Rel. max. at \((2,1)\) Concavity: Downward on \((-\infty, 0) \cup(0,3)\); upward on \((3, \infty)\) Point of inflection: \(\left(3, \frac{8}{9}\right)\)

Determine where the function is concave upward and where it is concave downward. $$ f(x)=\frac{1}{x-2} $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=x^{2} e^{-x} $$

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ h(x)=e^{x^{2}-4} \text { on }[-2,2] $$

Use the information summarized in the table to sketch the graph of \(f\). \(f(x)=x-3 x^{1 / 3}\) Domain: \((-\infty, \infty)\) Intercepts: \(x\) -intercepts: \(\pm 3 \sqrt{3}, 0\) Asymptotes: None Intervals where \(f\) is \(\nearrow\) and \(\searrow: \nearrow\) on \((-\infty,-1) \cup(1, \infty) ;\) \(\searrow\) on \((-1,1)\) Relative extrema: Rel. max. at \((-1,2) ;\) rel. min. at \((1,-2)\) Concavity: Downward on \((-\infty, 0)\); upward on \((0, \infty)\) Point of inflection: \((0,0)\)

Determine where the function is concave upward and where it is concave downward. $$ g(x)=\frac{x}{x+1} $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=e^{-x^{2} / 2} $$

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ g(x)=(2 x-1) e^{-x} \text { on }[0,4] $$

Sketch the graph of the function, using the curve-sketching quide of this section. $$ g(x)=4-3 x-2 x^{3} $$

Determine where the function is concave upward and where it is concave downward. $$ f(x)=\frac{1}{2+x^{2}} $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=\frac{\ln x}{x} $$

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=x e^{-x^{2}} \text { on }[0,2] $$

Sketch the graph of the function, using the curve-sketching quide of this section. $$ f(x)=x^{2}-2 x+3 $$

Determine where the function is concave upward and where it is concave downward. $$ g(x)=\frac{x}{1+x^{2}} $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=\ln x^{2} $$

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=x-\ln x \text { on }\left[\frac{1}{2}, 3\right] $$

Sketch the graph of the function, using the curve-sketching quide of this section. $$ h(x)=x^{3}-3 x+1 $$

Determine where the function is concave upward and where it is concave downward. $$ h(t)=\frac{t^{2}}{t-1} $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=\frac{x^{2}-1}{x} $$

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ g(x)=\frac{x}{\ln x} \text { on }[2,5] $$

Sketch the graph of the function, using the curve-sketching quide of this section. $$ f(x)=2 x^{3}+1 $$

Determine where the function is concave upward and where it is concave downward. $$ f(x)=\frac{x+1}{x-1} $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ h(x)=\frac{x^{2}}{x-1} $$

A stone is thrown straight up from the roof of an \(80-\mathrm{ft}\) building. The height (in feet) of the stone at any time \(t\) (in seconds), measured from the ground, is given by $$ h(t)=-16 t^{2}+64 t+80 $$ What is the maximum height the stone reaches?

Sketch the graph of the function, using the curve-sketching quide of this section. $$ f(x)=-2 x^{3}+3 x^{2}+12 x+2 $$

Determine where the function is concave upward and where it is concave downward. $$ g(x)=x+\frac{1}{x^{2}} $$

Lynbrook West, an apartment complex, has 100 two-bedroom units. The monthly profit (in dollars) realized from renting out \(x\) apartments is given by $$ P(x)=-10 x^{2}+1760 x-50,000 $$ To maximize the monthly rental profit, how many units should be rented out? What is the maximum monthly profit realizable?

Sketch the graph of the function, using the curve-sketching quide of this section. $$ f(t)=2 t^{3}-15 t^{2}+36 t-20 $$

Determine where the function is concave upward and where it is concave downward. $$ h(r)=-\frac{1}{(r-2)^{2}} $$

Sketch the graph of the function, using the curve-sketching quide of this section. $$ h(x)=\frac{3}{2} x^{4}-2 x^{3}-6 x^{2}+8 $$

Determine where the function is concave upward and where it is concave downward. $$ g(t)=(2 t-4)^{1 / 3} $$

The altitude (in feet) attained by a model rocket \(t\) sec into flight is given by the function $$ h(t)=-\frac{1}{3} t^{3}+4 t^{2}+20 t+2 \quad(t \geq 0) $$ Find the maximum altitude attained by the rocket.

Sketch the graph of the function, using the curve-sketching quide of this section. $$ f(t)=3 t^{4}+4 t^{3} $$

Determine where the function is concave upward and where it is concave downward. $$ f(x)=(x-2)^{2 / 3} $$

Data show that the number of nonfarm, full-time, self-employed women can be approximated by $$ N(t)=0.81 t-1.14 \sqrt{t}+1.53 \quad(0 \leq t \leq 6) $$ where \(N(t)\) is measured in millions and \(t\) is measured in 5 -yr intervals, with \(t=0\) corresponding to the beginning of 1963\. Determine the absolute extrema of the function \(N\) on the interval \([0,6]\). Interpret your results.

Sketch the graph of the function, using the curve-sketching quide of this section. $$ f(t)=\sqrt{t^{2}-4} $$

Determine where the function is concave upward and where it is concave downward. $$ f(x)=\frac{e^{x}-e^{-x}}{2} $$

AvERAGE SPEED OF A VEHICLE The average speed of a vehicle on a stretch of Route 134 between 6 a.m. and 10 a.m. on a typical weekday is approximated by the function $$ f(t)=20 t-40 \sqrt{t}+50 \quad(0 \leq t \leq 4) $$ where \(f(t)\) is measured in miles per hour and \(t\) is measured in hours, with \(t=0\) corresponding to 6 a.m. At what time of the morning commute is the traffic moving at the slowest rate? What is the average speed of a vehicle at that time?

Sketch the graph of the function, using the curve-sketching quide of this section. $$ f(x)=\sqrt{x^{2}+5} $$

Determine where the function is concave upward and where it is concave downward. $$ f(x)=x e^{x} $$

The management of Trappee and Sons, producers of the famous TexaPep hot sauce, estimate that their profit (in dollars) from the daily production and sale of \(x\) cases (each case consisting of 24 bottles) of the hot sauce is given by $$ P(x)=-0.000002 x^{3}+6 x-400 $$ What is the largest possible profit Trappee can make in 1 day?

Sketch the graph of the function, using the curve-sketching quide of this section. $$ g(x)=\frac{1}{2} x-\sqrt{x} $$

Determine where the function is concave upward and where it is concave downward. $$ f(x)=x^{2}+\ln x^{2} $$

The quantity demanded each month of the Walter Serkin recording of Beethoven's Moonlight Sonata, manufactured by Phonola Record Industries, is related to the price/compact disc. The equation $$ p=-0.00042 x+6 \quad(0 \leq x \leq 12,000) $$ where \(p\) denotes the unit price in dollars and \(x\) is the number of discs demanded, relates the demand to the price. The total monthly cost (in dollars) for pressing and packaging \(x\) copies of this classical recording is given by $$ C(x)=600+2 x-0.00002 x^{2} \quad(0 \leq x \leq 20,000) $$ To maximize its profits, how many copies should Phonola produce each month? Hint: The revenue is \(R(x)=p x\), and the profit is \(P(x)=\) \(R(x)-C(x)\).

Sketch the graph of the function, using the curve-sketching quide of this section. $$ f(x)=\sqrt[3]{x^{2}} $$

Determine where the function is concave upward and where it is concave downward. $$ f(x)=\frac{\ln x}{x} $$