6. Graphical Applications of Derivatives

Concavity

6. Graphical Applications of Derivatives

Concavity

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Multiple Choice

Determine the intervals for which the function is concave up or concave down. State the inflection points.
$f(x)=2x(x+9)^2$

Concave down: $\left(-\infty,\infty\right)$ ; No Inflection Points

Concave down: $\left(-\infty,-6\right)$ ; Concave up: $\left(-6,\infty\right)$ ; Inflection pt: $\left(-6,-108\right)$

Concave down: $\left(-\infty,6\right)$ ; Concave up: $\left(6,\infty\right)$ ; Inflection pt: $\left(6,2700\right)$

Concave down: $\left(-\infty,0\right)$ ; Concave up: $\left(0,\infty\right)$ ; Inflection pt: $\left(0,0\right)$

Verified step by step guidance

Step 1: To determine concavity, calculate the second derivative of the function f(x). Start by finding the first derivative f'(x) using the product rule and chain rule. The function is f(x) = 2x(x+9)^2.

Step 2: Differentiate f'(x) again to find the second derivative f''(x). Simplify the expression to make it easier to analyze the sign of f''(x).

Step 3: Set f''(x) = 0 to find the critical points where the concavity might change. Solve for x to determine these points.

Step 4: Use a test interval method or sign chart to analyze the sign of f''(x) in the intervals determined by the critical points. If f''(x) > 0, the function is concave up; if f''(x) < 0, the function is concave down.

Step 5: Identify the x-values where the concavity changes (from concave up to concave down or vice versa). These x-values are the inflection points. Substitute these x-values into the original function f(x) to find the corresponding y-coordinates of the inflection points.