Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.

ƒ(x) = x/(x²+9)⁵ on [-2,2]

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First, understand that absolute maxima and minima refer to the highest and lowest values of a function on a given interval. To find these, we need to evaluate the function at critical points and endpoints of the interval.

Find the derivative of the function ƒ(x) = x/(x²+9)⁵. Use the quotient rule, which states that if you have a function g(x)/h(x), its derivative is (g'(x)h(x) - g(x)h'(x))/(h(x))².

Set the derivative equal to zero to find critical points. This involves solving the equation derived from the derivative for x. Critical points occur where the derivative is zero or undefined.

Evaluate the function ƒ(x) at the critical points found in the previous step, as well as at the endpoints of the interval, x = -2 and x = 2.

Compare the values of ƒ(x) at these points to determine the absolute maximum and minimum values on the interval [-2, 2]. The largest value will be the absolute maximum, and the smallest will be the absolute minimum.

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주요 개념

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Absolute Extrema

Absolute extrema refer to the highest and lowest values of a function on a given interval. To find these values, one must evaluate the function at critical points, where the derivative is zero or undefined, as well as at the endpoints of the interval. The largest of these values is the absolute maximum, while the smallest is the absolute minimum.

Critical Points

Critical points are values of the independent variable where the derivative of the function is either zero or does not exist. These points are essential for finding absolute extrema, as they indicate where the function may change direction. To locate critical points, one must first compute the derivative of the function and solve for when it equals zero or is undefined.

Closed Interval Method

The closed interval method is a technique used to find absolute extrema of a function on a closed interval [a, b]. This method involves evaluating the function at the endpoints of the interval and at any critical points found within the interval. The absolute maximum and minimum values are then determined by comparing these function values.

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.ƒ(x) = x/(x²+9)⁵ on [-2,2]