5. Graphical Applications of Derivatives

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

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

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

$f(x)=x^{2/3}(4-x)$

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

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

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

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

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]

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√(4 - x²) on [-2,2]

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) = 2x⁵ - 5x⁴ - 10x³ + 4 on [-2,4]

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/(x² - 1) on [-4,4]

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?

Use ƒ' and ƒ" to complete parts (a) and (b). 

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

b. Find the intervals on which f is concave up and the intervals on which it is concave down.

ƒ(x) = x⁹/9 + 3x⁵ - 16x

Use ƒ' and ƒ" to complete parts (a) and (b). 

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

b. Find the intervals on which f is concave up and the intervals on which it is concave down.

ƒ(x) = x√(x +9)