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) = 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

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

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

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?

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?

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

f(x) = ³√(x - 4)

Let ƒ(x) = (x - 3) (x + 3)²

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

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

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

114. Parabolas

b. When is the parabola concave up? 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.

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

132. Let f(x) = e^x / (1 + e^(2x)).

b. Find all inflection points for f.