Question

Let f : ℝ → ℝ be a function such that:

|f(x) − f(y)| ≤ ½ |x − y|, for all x, y ∈ ℝ 

f′(x) ≥ ½, for all x ∈ ℝ, and f(1) = ½

Find the number of points of intersection of the curves:

y = f(x) and y = x² − 2x − 5

• A. 1
• B. 0
• C. 2
• D. Infinite

Hint:

A function both Lipschitz with slope ≤ k and derivative ≥ k implies constant slope. Use this to simplify and solve for intersections.

Answer 1

The Correct Option is C

Analysis 

From the given:

• |f(x) - f(y)| ≤ ½|x - y| ⇒ f is Lipschitz continuous, slope ≤ 0.5
• f′(x) ≥ ½ ⇒ Contradiction unless f′(x) = ½ ⇒ f is linear

So assume:

f(x) = (1/2)x + c

Use the point: f(1) = 1/2 ⇒ (1/2)·1 + c = 1/2 ⇒ c = 0

Therefore:

f(x) = (1/2)x

Intersection with y = x² − 2x − 5

Solve:

(1/2)x = x² − 2x − 5 
⇒ x² − (5/2)x − 5 = 0 
⇒ 2x² − 5x − 10 = 0 
⇒ x = [5 ± √(25 + 80)] / 4 
⇒ x = [5 ± √105] / 4

Therefore, there are 2 real points of intersection.