1. Optimization is done by finding the derivative of a function, then setting it equal to zero. The final step is to solve for the variable. We use the derivative and the process of optimization to find the maximum or minimum. For example,
2. Find the point on the on the line y=2x+3 that is closest to the origin. To optimize, use the distance formula and the original equation. d= sqrt[(x-0)^2 + (y-0)^2]
Plug in the original equation for y in the distance formula. d= sqrt[x^2 + (2x+3)^2]
Remove the square root and take the derivative of the equation. (d^2)' = 2x + 2*(2x+3)*2
Solve for x. (d^2)'= 2x +8x+12
Set the equation equal to zero to solve. 0 = 10x + 12 -12 = 10x x= -12/10 x= -6/5
Plug the x value back into y equals to find the point that is asked for in the original question and solve for y. y = 2(-6/5)+3 y = -12/5 + 3 y = 3/5
So the answer to the point that is closest to the origin is (-6/5, 3/5)
3. If f'(x) = 6x^2 -10x -1 then,
a)f(x) = 2x^3 -5x^2 -x + 7
b)f(x) = 2x^3 -5x^2 -x -15
