For the function, locate any absolute extreme points over the given interval. (Round your answers to three decimal places. If an answer does not exist, enter DNE.)g(x) = −3x2 + 14.7x − 16.7, −1 ≤ x ≤ 5
Question
For the function, locate any absolute extreme points over the given interval. (Round your answers to three decimal places. If an answer does not exist, enter DNE.)g(x) = −3x2 + 14.7x − 16.7, −1 ≤ x ≤ 5
Solution
To find the absolute extreme points of the function g(x) = -3x^2 + 14.7x - 16.7 over the interval -1 ≤ x ≤ 5, we need to follow these steps:
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Find the derivative of the function g(x). The derivative of g(x) is g'(x) = -6x + 14.7.
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Set the derivative equal to zero and solve for x to find critical points. So, -6x + 14.7 = 0. Solving for x gives x = 14.7/6 = 2.45.
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Evaluate the function at the critical points and at the endpoints of the interval. So, we need to find the values of g(-1), g(2.45), and g(5).
g(-1) = -3(-1)^2 + 14.7(-1) - 16.7 = -35.4 g(2.45) = -3(2.45)^2 + 14.7(2.45) - 16.7 = 0.725 g(5) = -3(5)^2 + 14.7(5) - 16.7 = -21.3
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The largest of these values is the absolute maximum, and the smallest is the absolute minimum. So, the absolute maximum is 0.725 at x = 2.45, and the absolute minimum is -35.4 at x = -1.
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