Final answer:
To find the critical numbers of the function y = x(16-x²)⁹⁄₂, differentiate the function and set the derivative to zero, then solve for x. Be sure to consider the domain of the original function and endpoints in your calculation.
Step-by-step explanation:
To find the critical number of the function y = x(16-x²)⁹⁄₂, we need to differentiate the function and set the derivative equal to zero to solve for x. Differentiating using the product rule and the chain rule will give us a derivative that we can set equal to 0 to find critical numbers.
Critical numbers occur where the derivative is zero or undefined. The critical numbers must be within the domain of the original function, which in this case is when x is between -4 and 4 since the term under the square root must be nonnegative. We may also need to check the endpoints of this interval to determine if they are critical numbers.
Applying the product rule, d/dx [x(16-x²)⁹⁄₂] involves computing the derivative of x and the derivative of the square root expression. After differentiation, set the derivative equal to zero and solve for x to find the critical numbers. If the equation is too complex to solve by common algebraic techniques, using the quadratic formula may be necessary when the derivative is of the form ax² + bx + c = 0.