5. Graphical Applications of Derivatives

Suppose the graph of $y=f\left(x\right)$ is shown above. Which of the following best describes the graph of $y=f^{″}\left(x\right)$?

At the point with x-coordinate $-2$, is the graph of $g\left(x\right)=4x^3+9x^2+7x+3$ concave up or concave down?

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

e. On what intervals (approximately) is f concave up?

Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.

f(x) = ³√(x - 4)

Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.

g(t) = 3t⁵ - 30t⁴ + 80t³ + 100

For the following graph, find the open intervals for which the function is concave up or concave down. Identify any inflection points.

For the following graph, find the open intervals for which the function is concave up or concave down. Identify any inflection points.

For the following graph, find the open intervals for which the function is concave up or concave down. Identify any inflection points.

The graph of $f^{\prime}\left(x\right)$is shown below. Use the graph to determine the intervals for which $f\left(x\right)$is concave up or concave down and the location of any inflection points.

The graph of $f''(x)$ is shown below. Use the graph to determine the intervals for which $f\left(x\right)$is concave up or concave down and the location of any inflection points.

Determine the intervals for which the function is concave up or concave down. State the inflection points.

$f(x)=2x(x+9)^2$

Determine the intervals for which the function is concave up or concave down. State the inflection points.

$f(x)=\frac{7}{2x-9}$

Determine the intervals for which the function is concave up or concave down. State the inflection points. 

$f(x)=2\sin x+3x$; $0$ < $x$ < $2\pi$

Concavity of parabolas Consider the general parabola described by the function f(x) = ax² + bx + c. For what values of a, b, and c is f concave up? For what values of a, b, and c is f concave down?

Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.

f(x) = 2x⁴ + 8x³ + 12x² - x - 2