Final answer:
To maximize xy² on the given ellipse, substitute y² in the expression with the equation of the ellipse, take the derivative, and find the value of x that maximizes the expression. The maximum value is 72.
Step-by-step explanation:
To maximize xy² on the ellipse 9x² + 16y² = 144, we need to find the maximum value of xy² that satisfies the equation.
First, we can solve the equation 9x² + 16y² = 144 for y² and rewrite it as y² = (144 - 9x²)/16.
Next, substitute y² in the expression xy² with the value we found: xy² = x((144 - 9x²)/16)².
To find the maximum value, we can take the derivative of xy² with respect to x and set it equal to zero. After solving, we find that x = 3/4. Substituting this value back into xy², we get the maximum value of (3/4)((144 - 9(3/4)²)/16)² = 72.
So the maximum value of xy² on the given ellipse is 72.