Among all pairs of numbers (x,y) such that 3x+y=15, find the minimum of x^2 + y^2.
+9466 Solve 3x + y = 15 for y to get y = 15 - 3x
Let
z(x) = x2 + y2 = x2 + (15 - 3x)2
So we want to find what value of x minimizes z(x)
First let's find values of x which make z'(x) = 0
z'(x) = 2x + 2(15 - 3x)(-3) = 2x - 90 + 18x = 20x - 90 = 0 ⇒ x = 4.5
Now let's check whether the graph is concave up or down when x = 4.5 using the second derivative test:
z''(x) = 20 ⇒ z''(4.5) = 20 > 0 so the graph is concave up
Since z'(4.5) = 0 and z''(4.5) > 0 , we can say a min occurs when x = 4.5
When x = 4.5, y = 15 - 3(4.5) = 1.5
And x2 + y2 = (4.5)2 + (1.5)2 = 22.5
