Optimization: sum of squares  Applications of derivatives  AP Calculus AB  Khan Academy  Summary and Q&A
TL;DR
The smallest sum of squares of two numbers with a negative product of 16 is 32.
Questions & Answers
Q: How can we find the smallest sum of squares of two numbers with a negative product of 16?
We can express the sum of squares in terms of x and y, and given their product is 16, we can substitute y with 16/x. By minimizing this expression, we find the smallest sum of squares.
Q: What is the derivative of the sum of squares function?
The derivative of the sum of squares with respect to x is 2x*(2)2x + 256(2). Simplifying, we get 512x^(3).
Q: What are the critical points of the sum of squares function?
The only critical point is x=4, since x=0 would result in an undefined y. This critical point minimizes the sum of squares.
Q: How can we confirm that x=4 results in a minimum value?
We can use the second derivative test. Taking the second derivative of the sum of squares function, we find that it is always positive, indicating a concave upwards shape and confirming the minimum point at x=4.
Summary & Key Takeaways

The goal is to find the smallest sum of squares of two numbers whose product is negative 16.

By expressing the sum of squares as a function of x and y, and using the given information about their product, we can rewrite the expression in terms of x only.

Taking the derivative of the function and finding the critical points, we find that the only critical point is x=4, which minimizes the sum of squares.

The minimum sum of squares is 32, with x=4 and y=4.