I will help with first two problems, but before this here is the link to get a better touch of optimization problems.
1.P is a moveable point on the line y=7-x. Find the coordinates of P when it is closest to the point (2,4).
Since P is a point on line y=7-x, then coordinates of point P are (x,7-x).
Now, calculate distance between point P and point (2,4):
Clearly if we find minimum of d then it will also be minimum for . Therefore, for simplicity we will find minimum of
Set f'(x)=0: 4x-10=0 or x=2.5.
Therefore, the closest point is P=(2.5,4.5).
2.A small boat moving at x km/h uses fuel at a rate that is approximated by the function q=8+x^2/50, where q is measured in litres/hour. Determine the speech of the boat for which the amount of fuel used for any given journey is least.
This means that if boat is moving at x km/h then it uses litres per kilometer.
We must minimize this quantity:
Set f'(x)=0: or or x=20 and x=-20. x=-20 is impossible because speed cannot be negative.
Therefore, only x=20 is applicable. Now we need to check that x=20 is indeed minimum.
For this we will use second derivative test: since then therefore x=20 is indeed minimum.
Answer: x=20 kilometers per hour.