5. Graphical Applications of Derivatives

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 .

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?

Does ƒ(x) = (x⁶/2) + (5x⁴/4) - 15x² have any inflection points? If so, identify them.

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

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

113. If b, c, and d are constants, for what value of b will the curve y = x^3 + bx^2 + cx + d have a

point of inflection at x = 1? Give reasons for your answer.

114. Parabolas

b. When is the parabola concave up? Concave down?

117. Suppose that the second derivative of the function y = f(x) isy" =(x+1)(x-2).

For what x-values does the graph of f have an inflection point?

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".

Each of Exercises 89–92 shows the graphs of the first and second derivatives of a function y=f(x). Copy the picture and add to it a sketch of the approximate graph of f, given that the graph passes through the point P.

103. A function f(x) has domain (-2, 2). The graph below is a plot of the derivative of f, not a plot of f itself. In other words, this is a graph of y = f'(x). Either use this graph to determine on which intervals the graph of f is concave up and on which intervals the graph of f is concave down, or explain why this information cannot be determined from the graph.

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?

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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