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

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To find inflection points, we need to determine where the second derivative of the function changes sign. Start by finding the first derivative of the function ƒ(x) = (x⁶/2) + (5x⁴/4) - 15x².

Differentiate ƒ(x) to get the first derivative ƒ'(x). Use the power rule: ƒ'(x) = d/dx[(x⁶/2) + (5x⁴/4) - 15x²].

Calculate the first derivative: ƒ'(x) = (6x⁵/2) + (5 * 4x³/4) - 30x = 3x⁵ + 5x³ - 30x.

Now, find the second derivative ƒ''(x) by differentiating ƒ'(x): ƒ''(x) = d/dx[3x⁵ + 5x³ - 30x].

Calculate the second derivative: ƒ''(x) = 15x⁴ + 15x² - 30. Set ƒ''(x) = 0 and solve for x to find potential inflection points. Check the sign change around these points to confirm inflection points.

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Inflection Points

Inflection points are points on a curve where the concavity changes, meaning the curve transitions from being concave up to concave down or vice versa. To find inflection points, one must analyze the second derivative of the function. If the second derivative changes sign at a certain point, that point is classified as an inflection point.

Second Derivative Test

The second derivative test involves taking the second derivative of a function to determine its concavity. If the second derivative is positive, the function is concave up; if negative, it is concave down. This test is crucial for identifying inflection points, as a change in the sign of the second derivative indicates a potential inflection point.

Critical Points

Critical points are values of x where the first derivative of a function is either zero or undefined. These points are important because they can indicate local maxima, minima, or inflection points. To find inflection points, one must first identify critical points and then analyze the second derivative at these points to check for changes in concavity.