Question

a) Given 2 functions f(x)= x^2 and g(x)= x^3 - x^2. At the points where the

two functions intersect, there exists a common tangent at only one of these
points. Determine the coordinates of the point where the common tangent exists.

Note : only calculus concepts learned in rate of change can be used

b) At the “other” x value where there is no common tangent, determine the
equation of the tangent at this point to the curve y = x^3 - x^2

answer and elaborate, please, I'm confused

Answer 1

didn't we just do one like this, with x^4 and x^5-x^4?

The curves for f and g intersect at (0,0) and (2,4)
f' and g' are both zero at (0,0), so they have a common tangent there.
f'(2) = 4
g'(2) = 8
so there is no common tangent there. Since we have a point and a slope, the tangent line for g(x) at (2,4) is
y-4 = 8(x-2)