# Find the point on the line y = 4x + 5 that is closest to the origin.

Find the point on the line y = 4x + 5 that is closest to the

We want to find the point on the line y = 4x +5 closest to the
origin (0,0). Lets use (x,y) for the point we want to find. Using
the distance formula:

We know that y = 4x + 5 so substituting this into the above
result yields:

We want to minimize the distance function so lets take the first
derivative of D:

To find the critical point(s) we set D' = 0 and solve for x:

is the critical point.

For values of x less than the critical point the sign of D' is
negative and for values of x greater than our critical point the
sign of D' is positive. The point:

is a local minimum by the First Derivative
Test.

We can use the second derivative test to verify this.

Now D" is positive for all values of x and when D'(CP) = 0 and
D"(CP) > 0 then there is a local minimum at the point
(CP,f(CP).

The point on the line y = 4x + 5 that is closest to the origin
is:

(-1.18,0.294)