Let the sum of two numbers be 15. use calculus to find the values of the numbers which make the sum of their squares a minimum.

Let the sum of the two numbers by x + y = 15, where x and y are the numbers.then the sum of their squares can be denoted by     s = x² + y²note that we have the equation x + y = 15 from before, which allows us to rewrite the sum in terms of x only. this allows us to apply techniques for single variable calculus.  we take x + y = 15 and subtract x from both sides to get y = 15 - x.then we can substitute this into the sum of their squares:     s(x) = x² + (15 - x)²the sum then becomes a function of x.to find the minimum, we need to find the critical points of function s.critical points are where s is negative or undefined.in order to do that, we need to differentiate the function s with respect to x: use the power rule and chain rule to differentiate:     s'(x) = 2x + 2(15 - x)· (d/dx)(15 - x)    s'(x) = 2x + 2(15 - )    s'(x) = 2x - 2(15 - x)    s'(x) = 2x - 30 + 2x    s'(x) = 4x - 30the derivative of function s is s'(x) = 4x - 30.find critical points by setting s'(x) to zero:     0 = 4x - 30    4x = 30    x = 30/4    x = 7.5x = 7.5 is a critical point of s'.we can verify that this is a relative minimum by using the first derivative test.set up two intervals using our critical points:     x < 7.5 and x > 7.5test the sign of f' in the two intervals using any value in those respective intervals.for interval x < 7.5, we can use x = 0 in s' to get    s'(0) = 4(0) - 30 = -30 < 0in  interval  x < 7.5, s' is negativefor  interval x > 7.5, we can use x = 8 in s' to get    s'(8) = 4(8) - 30 = 32 - 30 = 2  > 0in  interval  x > 7.5, s' is positive.the derivative changes from negative to positive at x = 7.5. therefore, this is a relative minimum.it is an absolute minimum because the shape of the graph of s(x) = x² + (15 - x)² is parabolic.therefore, x = 7.5 results in a minimum. one of the numbers is 7.5find the other number,y,  using the equation x + y = 15  ⇒ y = 15 - x    y = 15 - 7.5 = 7.5hence the two numbers are both 7.5