Q:

The graphs of y equals 1.1 Superscript x and yequalsx have two points of​ intersection, while the graphs of y equals 2 Superscript x and yequalsx have no points of intersection. It follows that there is a real number 1 less than p less than 2 for which the graphs of y equals p Superscript x and yequalsx have exactly one point of intersection and yequalsx is tangent to y equals p Superscript x. Using analytical​ and/or graphical​ methods, determine p and the coordinates of the single point of intersection.

Accepted Solution

A:
Answer:p = 1The coordinates where the curves intersect is (1, 1).Step-by-step explanation:Hi there!We Know that y = x is tangent to y = pˣ, then the derivative of y = p× is y = x:[tex]y = p^{x}[/tex][tex]dy/dx = xp^{x-1}[/tex]Since dy/dx = y = x[tex]x = xp^{x-1}\\1 = p^{x-1}\\ln(1) = ln(p^{x-1})\\0 = (x-1)ln(p)\\0 = ln(p)\\p = 1[/tex]Then, if p = 1, the function will be:y = 1ˣ Let´s find the intersection with the function y = x. We have the following system of equations:y = xy = 1ˣ Replacing y in the second equation:x = 1ˣApply ln to both sides of the equation:ln (x) = ln(1ˣ)Apply logarithm property: ln(xᵃ) = a ln(x)ln(x) = x ln(1)ln(x) = 0Apply e to both sides of the equation:e^(ln(x)) = e^0x = 1then:y = 1The coordinates where the curves intersect is (1, 1)See the attached figure for a graphical representation:in blue: y = 1ˣin red: y = xHave a nice day!