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

Cubic functions Consider the cubic function f(x) = ax³ + bx² + cx + d.

b. How many local extreme values can f have?

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.

g(x) = x√8 − x²

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)

Identifying Extrema

In Exercises 41–52:

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

f(x) = √(x² − 2x − 3), 3 ≤ x < ∞

In Exercises 121–124, find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of the function’s first and second derivatives. How are the values at which these graphs intersect the x-axis related to the graph of the function? In what other ways are the graphs of the derivatives related to the graph of the function?

123. y=(4/5)x^5+16x^2-25

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 $[0,\pi]$