5. Graphical Applications of Derivatives

Points of inflection Find the x-coordinate of the point(s) of inflection of f(x) = tanh² x.

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

107. Marginal cost The accompanying graph shows the hypothetical cost c=f(x) of manufacturing x items. At approximately what production level does the marginal cost change from decreasing to increasing?

99. In Exercises 99 and 100, the graph of f' is given. Determine x-values corresponding to inflection points for the graph of f.

Let ƒ(x) = (x - 3) (x + 3)²

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

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

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

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

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.

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.

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]

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

f. On what intervals (approximately) is f concave down?  

Graph f(x) = x cos x and its second derivative together for 0 ≤ x ≤ 2pi. Comment on the behavior of the graph of f in relation to the signs and values of f".

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

c. At what point(s) does f have an inflection point?

93. The accompanying figure shows a portion of the graph of a twice-differentiable function y=f(x). At each of the five labeled points, classify y' and \(\y\)'' as positive, negative, or zero.

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