Hello I need step by step all way to get the answer.
1) Find the closest distance of the point (3,1) to the curve xy=8.
2) Find the farthest distance of the point (3,0) to the curve xy=8.
Have you been taught how to solve an optimisation problem?
You find the stationary points and then check the nature of these stationary points. If you want to maximise then you keep the solution that is a maximum. If you want to minimise then you keep the solution that is a minimum.
I have set it up for you! All that is essentially left is to find the coordinates of the maximum turning point of D^2. Surely you can do this? If not, you need to go back to your classnotes and textbook and thoroughly review the material that this topic is based on. To find the derivative, I suggest you apply some basic algebra and expand the expression I gave you. Then differentiate it term-by-term.
If you want more help, show some effort. We are not going to just hand you a step-by-step by solution.
Your objective is to find the derivative of $\displaystyle (a-3)^{2} + \left( \frac{8}{a} - 1\right)^2$ and set it equal to 0.
Taking each part in turn:
Step1
Here is the differenciation of $\displaystyle f(a) = \left( \frac{8}{a} - 1\right)^2$
rewrite this as
$\displaystyle f(a) = \left( 8a^{-1} - 1\right)^2$
From here, just differentiate using the chain rule:
$\displaystyle f'(a) = 2 \left( 8a^{-1} - 1\right)^{1} * -8a^{-2} $
$\displaystyle f'(a) = -16a^{-2} \left( 8a^{-1} - 1\right)$
In case your chain rule is a bit rusty, it goes like this:
Bring the power of the bracket (2) down and reduce the power on the bracket by 1.
Then multiply by the deriviative of the bracket
Step2
Now, we need the derivative of:
$\displaystyle (a-3)^{2}$
which is $\displaystyle 2(a-3)$
Step3
Combining those 2 results, we have the derivative we want. Your optimisation problem is:
$\displaystyle 2(a-3) + -16a^{-2} \left( 8a^{-1} - 1\right) =0 $
Have a go at that. I might have made a mistake because that equation looks pretty messy to solve.
You are expected to know how to expand and simplify the expression I gave for D^2:
$\displaystyle D^2 = a^2 - 6a - \frac{16}{a} + \frac{64}{a^2} + 10$.
You are expecetd to know how to differentiate each term in the above:
$\displaystyle \frac{d \, D^2}{da} = 2a - 6 + \frac{16}{a^2} - \frac{128}{a^3}$.