# Math Help - Optimazation Problems

1. ## Optimazation Problems

This one is killing me:

Find the point on the line y = 4x + 7 that is closest to the origin.

I keep trying the distance formula and I end up with:
d = [ 17x^2 + 56x + 49]^1/2
because I substituted for y, getting it down to two variables like my teacher said. The solutions manual keeps saying I'm wrong because after I differentiate that I get a different answer. They do something weird where they square it and then differentiate to make it easier to work with. I get an ugly looking fraction when I differentiate my findings.

2. $D = \sqrt{17x^2 +56x + 49}$

$\dfrac{dD}{dx} = \dfrac12(17x^2 + 56x + 49)^{-\frac12}. (34x + 56)$

Set D' to 0.

$0 = (17x + 28)(17x^2 + 56x + 49)^{-\frac12}$

$x = -\dfrac{28}{17}$

$y = 4\left(-\dfrac{28}{17}\right) + 7 = \dfrac{7}{17}$

Are you getting this?