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Find the maximum value of f(x,y)=x³y^7 on the unit circle, where x and y are greater than or equal to 0.

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Final answer:

To find the maximum value of f(x, y) on the unit circle in the first quadrant, substitute y with √(1 - x²), reduce to a single variable function, find its derivative, critical points, and evaluate the function at these points and interval endpoints.

Step-by-step explanation:

The student is asking to find the maximum value of the function f(x, y) = x³y⁷ on the unit circle x² + y² = 1, where x and y are both non-negative. To find this maximum, since we're constrained to the unit circle, we can substitute y with √(1 - x²). This gives us a function of a single variable f(x) = x³(1 - x²)³⁷/2. However, because x and y are non-negative, we only consider the first quadrant of the unit circle, where 0 ≤ x ≤ 1. To find the maximum value, we can then take the derivative of f(x) with respect to x and find the critical points within the closed interval [0,1]. After finding the critical points, we can use them to find the maximum value of f(x) by evaluating the function at these points and at the endpoints of the interval.

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