미분 전체 문제

For the following exercises, determine

a. intervals where $f$ is increasing or decreasing, b. local minima and maxima of $f,$ c. intervals where $f$ is concave up and concave down, and d. the inflection points of $f.$

$f(x)={x}^{3}+{x}^{4}$

For the following exercises, determine

a. intervals where $f$ is increasing or decreasing, b. local minima and maxima of $f,$ c. intervals where $f$ is concave up and concave down, and d. the inflection points of $f.$ Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator.

[T] $f(x)=\sin (\pi x)-\cos (\pi x)$ over $x=[-1,1]$

For the following exercises, determine

a. intervals where $f$ is increasing or decreasing, b. local minima and maxima of $f,$ c. intervals where $f$ is concave up and concave down, and d. the inflection points of $f.$ Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator.

[T] $f(x)=x+\sin (2x)$ over $x=\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$

For the following exercises, determine

a. intervals where $f$ is increasing or decreasing, b. local minima and maxima of $f,$ c. intervals where $f$ is concave up and concave down, and d. the inflection points of $f.$ Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator.

[T] $f(x)=\sin x+\tan x$ over $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$

For the following exercises, determine

a. intervals where $f$ is increasing or decreasing, b. local minima and maxima of $f,$ c. intervals where $f$ is concave up and concave down, and d. the inflection points of $f.$ Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator.

[T] $f(x)={(x-2)}^{2}{(x-4)}^{2}$

For the following exercises, determine

a. intervals where $f$ is increasing or decreasing, b. local minima and maxima of $f,$ c. intervals where $f$ is concave up and concave down, and d. the inflection points of $f.$ Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator.

[T] $f(x)=\frac{\sin x}{x}$ over $x=$ $[2\pi ,0)\cup (0,2\pi ]$

For the following exercises, determine

a. intervals where $f$ is increasing or decreasing, b. local minima and maxima of $f,$ c. intervals where $f$ is concave up and concave down, and d. the inflection points of $f.$ Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator.

$f(x)=\sin (x){e}^{x}$ over $x=[-\pi ,\pi ]$

For the following exercises, determine

a. intervals where $f$ is increasing or decreasing, b. local minima and maxima of $f,$ c. intervals where $f$ is concave up and concave down, and d. the inflection points of $f.$ Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator.

$f(x)=(\ln x)\sqrt{x},x>0$

For the following exercises, determine

a. intervals where $f$ is increasing or decreasing, b. local minima and maxima of $f,$ c. intervals where $f$ is concave up and concave down, and d. the inflection points of $f.$ Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator.

$f(x)=\frac{1}{4}\sqrt{x}+\frac{1}{x},x>0$

For the following exercises, determine

a. intervals where $f$ is increasing or decreasing, b. local minima and maxima of $f,$ c. intervals where $f$ is concave up and concave down, and d. the inflection points of $f.$ Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator.

$f(x)=\frac{{e}^{x}}{x},x\ne 0$

The population is growing more slowly. Here $f$ is the population.

A bike accelerates faster, but a car goes faster. Here $f=$ Bike’s position minus Car’s position.

The airplane lands smoothly. Here $f$ is the plane’s altitude.

A company tracks labor costs over the course of a year. On July 1, costs are at their peak. Here $f$ represents the cost of labor.

The economy is picking up speed. Here $f$ is a measure of the economy, such as GDP.

For the following exercises, consider a third-degree polynomial $f(x),$ which has the properties ${f}^{'}(1)=0,{f}^{'}(3)=0$ (not applicable to Exercises 249 and 250). Determine whether the following statements are true or false. Justify your answer.

$f(x)=0$ for some $1\le x\le 3$

For the following exercises, consider a third-degree polynomial $f(x),$ which has the properties ${f}^{'}(1)=0,{f}^{'}(3)=0$ (not applicable to Exercises 249 and 250). Determine whether the following statements are true or false. Justify your answer.

${f}^{''}(x)=0$ for some $1\le x\le 3$

For the following exercises, consider a third-degree polynomial $f(x),$ which has the properties ${f}^{'}(1)=0,{f}^{'}(3)=0$ (not applicable to Exercises 249 and 250). Determine whether the following statements are true or false. Justify your answer.

There is no absolute maximum at $x=3$

If $f(x)$ has one inflection point, then it has three real roots.

When you find the maximum for an optimization problem, why do you need to check the sign of the derivative around the critical points?