a) Given 2 functions f(x)= x^4 and g(x)= x^5 - x^4. At the points where the

Question

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^5 - x^4

Answer 1

Intersection points:
2 x^4 = x^5
x^5 - 2 x^4
x^4 (x-2) = 0
x = 0 or x = 2
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tangent slopes
f' = 4 x^3
g' = 5 x^4 - 4 x^3
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at x = 0 both slopes are zero and both f and g are zero
at x = 2, f' = 32 and g' = 5(16) - 4(8) = 80-32 = 48 =slope m
so x = 2 , y = 32-16 = 16 and slope = 48
y = m x + b
16 = 48 (2) + b
16 = 96 + b
b = -80
so
y = 48 x - 80