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.

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mauricioalessandrell mauricioalessandrell · Answer 1 · 2019-11-11T16:16:48+01:00

Answer:

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:

[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 = 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!