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

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To determine concavity, we need to find the second derivative of the function f(x). Start by finding the first derivative f'(x) of the function f(x) = 2x⁴ + 8x³ + 12x² - x - 2.

Calculate the first derivative: f'(x) = d/dx [2x⁴ + 8x³ + 12x² - x - 2]. Use the power rule for differentiation: f'(x) = 8x³ + 24x² + 24x - 1.

Next, find the second derivative f''(x) by differentiating f'(x): f''(x) = d/dx [8x³ + 24x² + 24x - 1]. Again, apply the power rule: f''(x) = 24x² + 48x + 24.

To find intervals of concavity, solve f''(x) = 0 to find critical points. Set 24x² + 48x + 24 = 0 and solve for x. This will give potential inflection points where concavity changes.

Use the critical points to test intervals on the number line. Choose test points in each interval and evaluate the sign of f''(x) at those points. If f''(x) > 0, the function is concave up; if f''(x) < 0, it is concave down. Identify any inflection points where the concavity changes.

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Concavity

Concavity refers to the direction in which a function curves. A function is concave up on an interval if its second derivative is positive, indicating that the slope of the tangent line is increasing. Conversely, it is concave down if the second derivative is negative, meaning the slope is decreasing. Understanding concavity helps in analyzing the behavior of functions and identifying intervals of increase or decrease.

Second Derivative Test

The second derivative test is a method used to determine the concavity of a function and locate inflection points. By calculating the second derivative of a function, we can assess where it changes sign. If the second derivative is positive, the function is concave up; if negative, it is concave down. Inflection points occur where the second derivative equals zero or is undefined, indicating a change in concavity.

Inflection Points

Inflection points are specific points on a curve where the concavity changes. These points are significant because they indicate a transition in the behavior of the function, which can affect its graph and the nature of its extrema. To find inflection points, one must solve for values of x where the second derivative is zero or undefined, and then verify that the concavity changes at those points.

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