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.

p = 1

The 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:

Since dy/dx = y = x

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 = x

y = 1ˣ

Replacing y in the second equation:

x = 1ˣ

Apply 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) = 0

Apply e to both sides of the equation:

e^(ln(x)) = e^0

x = 1

then:

y = 1

The coordinates where the curves intersect is (1, 1)

See the attached figure for a graphical representation:

in blue: y = 1ˣ

in red: y = x

Have a nice day!

could you give more information?