First Question:
Let curve 1 be:
Let curve 2 be:
Thus, the curves intersect where:
is extraneous, since
has no solution for this
-value
when
:
So
is the ONLY point of intersection. Now let's find the slopes of the tangent lines of each curve at this point. If the slopes are perpendicular, that is, one is the negative inverse of the other, then we have that the curves are orthogonal.
For curve 1:
at
,
For curve 2:
at
,
Thus we see that the curves are orthogonal, since -2 is the negative reciprocal of
Question 14:
By the product rule:
I skipped a lot of steps, i don't think you'll have a problem getting here.
This is a downward opening parabola, the derivative is maximum at it's vertex. That is,
is max at
So we have that the tangent of greatest slope occurs at
and it's value is 3. I think you can take it from here and find the equation of the tangent line