5. Graphical Applications of Derivatives

Theory and Examples

Maximum height of a vertically moving body The height of a body moving vertically is given by s = −12gt² + υ₀t + s₀,  g > 0, with s in meters and t in seconds. Find the body’s maximum height.

Finding Critical Points

In Exercises 41–50, determine all critical points and all domain endpoints for each function.

y = x − 3x²ᐟ³

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) = x³ / (3x² + 1)

Sketch the graph of a twice-differentiable function y=f(x) that passes through the points (-2,2), (-1,1), (0,0),(1,1), and (2,2) and whose first two derivatives have the following sign patterns.

Theory and Examples

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

Let f(x) = (x − 2)²ᐟ³.

a. Does f′(2) exist?

Finding Critical Points

In Exercises 41–50, determine all critical points and all domain endpoints for each function.

y = x² + 2/x

First Derivative Test

a. Locate the critical points of f.

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).

f(x) = x² + 3 on [-3,2]

Finding Critical Points

In Exercises 41–50, determine all critical points and all domain endpoints for each function.

f(x) = x(4 − x)³

Identify the open intervals on which the function is increasing or decreasing.

$f(x)=3x^4+8x^3-18x^2+7$

Identify the local minimum and maximum values of the given function, if any.

$f\(\left\)(\(\theta\[\right\))=\(\sin\]\theta\)+\(\cos\)^2\(\theta\)$ on $[0,π]$

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

g(x) = x(ln x)²

Identify the intervals on which the function is increasing or decreasing.

$f(x)=\(\sin\)^2x$ on $[0,\(\pi\)]$

First Derivative Test

a. Locate the critical points of f.

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).

f(x) = x - 2 tan⁻¹ x on [-√3,√3)

Interpreting the derivative The graph of f' on the interval [-3,2] is shown in the figure. <IMAGE>

a. On what interval(s) is f increasing? Decreasing?

Theory and Examples

In Exercises 53 and 54, show that the function has neither an absolute minimum nor an absolute maximum on its natural domain.

y = x¹¹ + x³ + x − 5