5. Graphical Applications of Derivatives

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

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

h(t) = 2 + cos 2t on [0,π]

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

f(x) = x⁴eˣ + x

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

f(x) = -x⁴ - 2x³ + 12x²

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

b. Determine the locations of the inflection points of f and the intervals on which f is concave up or concave down.

Let ƒ(x) = (x - 3) (x + 3)²

d. Determine the intervals on which ƒ is concave up or concave down.

Let ƒ(x) = (x - 3) (x + 3)²

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