hey, can anybody help solve this HW question? thanks!
Write a function that models the distance D from a point on the line y = 3 x - 6 to the point (0,0) (as a function of x).
Each point on the line can be expressed as a coordinate in the form...
$\displaystyle (x, 3x - 6)$.
Why? A coordinate is of the form $\displaystyle (x,y)$ and $\displaystyle y = 3x - 6$
So what to do is think in terms of Pythagoras triangle... You have two sides, one of length $\displaystyle x$, one of length $\displaystyle 3x-6$ and one unknown side that you must find.
Why? Well just think of the coordinate $\displaystyle (x, 3x - 6)$ as being distance along the x-axis then distance up to get to the point. This will form a right angled triangle with side lengths $\displaystyle x$ and $\displaystyle 3x -6$
So, let's call this unknown side $\displaystyle d$, using the standard Pythagoras formula we get...
$\displaystyle d^2 = x^2 + (3x - 6)^2.$
Now you finish this off by expanding everything and you will be left with a formula of the form...
$\displaystyle d^2 = ...$
then just take square roots to get,
$\displaystyle d = \sqrt{\dots}$
which will be your answer.
Well you want to find the value of $\displaystyle x$ that makes that function the smallest.
This will happen at one of two places.
A critical point,
An end point of the range of $\displaystyle x$. (i.e. $\displaystyle x = \pm \infty$).
Immediately you can see that that if $\displaystyle x \to \pm \infty$ then $\displaystyle f(x) \to \infty$ (if not I'll explain).
So that leaves you to find critical points...
I'll start you off...
If $\displaystyle f(x) = \sqrt{10x^2 - 36x + 36}$
=> $\displaystyle f'(x) = \frac{2(5x-9)}{\sqrt{10x^2 - 36x + 36}}$.
Solve for this to be 0 and what does that tell you..?