5. Graphical Applications of Derivatives

Identifying Extrema

In Exercises 41–52:

a. Identify the function’s local extreme values in the given domain, and say where they occur.

g(x) = −x² − 6x − 9,−4 ≤ x < ∞

Identifying Extrema

In Exercises 19–40:

a. Find the open intervals on which the function is increasing and those on which it is decreasing.

b. Identify the function’s local extreme values, if any, saying where they occur.

f(x) = x¹ᐟ³(x + 8)

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.

__________

y = √ 3 + 2𝓍 ―𝓍²

Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. <IMAGE>

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.

y = (𝓍 + 1) / (𝓍² + 2𝓍 + 2)

Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>

c. Determine where f has local maxima and minima.

{Use of Tech} Spring runoff The flow of a small stream is monitored for 90 days between May 1 and August 1. The total water that flows past a gauging station is given by v(t) = <matrix 2x2> where V is measured in cubic feet and t is measured in days, with t=0 corresponding to May 1.

c. Describe the flow of the stream over the 3-month period. Specifically, when is the flow rate a maximum?

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.

f(x) = x² - 2 ln x

Which of the following best explains why the function $f\left(x\right)=\frac{1}{x-2}$ is discontinuous at $a=2$?

The position function of a particle is given by $s\left(t\right)=t^3-6t^2+9t+2$. At what time $t$ is the speed of the particle minimum?

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.

ƒ(x) = 1 / x + ln x

Which of the following could be a turning point for the continuous function $f\left(x\right)$?

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.

ƒ(x) = x √(x-a)

Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>

Let the function be defined by $f\left(x\right)=x^3-3x^2+2$. At what value(s) of $x$ does $f\left(x\right)$ have a relative maximum?