5. Graphical Applications of Derivatives

Concavity

5. Graphical Applications of Derivatives

Concavity

Struggling with Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

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

First, find the second derivative of the function f(x) = 2x(x+9)^2. Start by expanding the function to make differentiation easier.

Differentiate the expanded function to find the first derivative, f'(x). Use the product rule and chain rule as necessary.

Differentiate f'(x) to find the second derivative, f''(x). This will help determine the concavity of the function.

Set the second derivative, f''(x), equal to zero and solve for x to find potential inflection points. These are the points where the concavity might change.

Analyze the sign of f''(x) on the intervals determined by the critical points found in the previous step. If f''(x) > 0, the function is concave up on that interval; if f''(x) < 0, the function is concave down. Identify the intervals of concavity and the inflection points.