5. Graphical Applications of Derivatives

For the curve $y=7\ln \left(x\right)$, at what value of $x$ does the curve have maximum curvature?

Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>

b. Give the approximate coordinates of the absolute maximum and minimum values of ƒ (if they exist).

if ƒ(x) = 1 / (3x⁴ + 5) , it can be shown that ƒ'(x) = 12x³ / (3x⁴ + 5)² and ƒ"(x) = 180x² (x² + 1) (x + 1) (x - 1) / (3x⁴ + 5)³ . Use these functions to complete the following steps.

d. Identify the local extreme values and inflection points of ƒ .

Let $f\left(x\right)$ = $\left\{\begin{array}{cc}ax+b& \text{,}x<2\\ x^2& \text{,}x\ge 2\end{array}\right\}$. For which values of $a$ and $b$ is $f\left(x\right)$ continuous everywhere?

Identifying Extrema

In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.

Consider the graph of $f^{\prime }\left(x\right)$ below. How many local maxima does $f\left(x\right)$ have?

Identifying Extrema

In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.

Identifying Extrema

In Exercises 53–60:

a. Find the local extrema of each function on the given interval, and say where they occur.

f(x) = sec²x − 2tan x, −π/2 < x < π/2

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

ƒ(x) = 3x² - 4x + 2

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

f(x) = x²/₃ (x²-4)

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

Theory and Examples

In Exercises 51 and 52, give reasons for your answers.

Let f(x) = |x³ − 9x|.

d. Determine all extrema of f.

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

a. Find the critical points of f and determine where f is increasing and where it is decreasing.

Which of the following statements is true about the absolute maximum and minimum values of a continuous function $f\left(x\right)$ on a closed interval $[a,b]$?

In the context of extrema, if all the rates of change $f^{\prime }\left(x\right)$ (derivatives) in a set of problems are negative, what does this indicate about the behavior of the functions involved?