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Question 1178643: Functions f(x) =1-|x-1| and g(x)=(2x-a) are given .prove that for every 'a' belong to (1,2).the area bounded by the graphs of the given function is less than 1/3.(prove without differentiation and integrals).
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! This is a challenging problem that requires geometric reasoning and careful consideration of the function graphs. Let's break down the proof:
**1. Understand the Graphs:**
* **f(x) = 1 - |x - 1|:**
* This is an absolute value function, creating a "V" shape.
* The vertex of the "V" is at (1, 1).
* The graph intersects the x-axis at x = 0 and x = 2.
* The graph is symmetric about the line x = 1.
* **g(x) = 2x - a:**
* This is a linear function with a slope of 2.
* The y-intercept is -a.
* Since a ∈ (1, 2), the y-intercept is between -2 and -1.
**2. Visualize the Bounded Area:**
* The graphs of f(x) and g(x) will intersect at two points, creating a bounded area.
* The shape of this bounded area will be a triangle.
**3. Find the Intersection Points:**
* To find the intersection points, set f(x) = g(x):
* 1 - |x - 1| = 2x - a
* We need to consider two cases for the absolute value:
* **Case 1: x ≥ 1**
* 1 - (x - 1) = 2x - a
* 2 - x = 2x - a
* 3x = 2 + a
* x = (2 + a) / 3
* **Case 2: x < 1**
* 1 - (1-x) = 2x -a
* x = 2x -a
* x = a
* Because a is within the interval (1,2) then the intersection points are x = a and x = (2+a)/3.
**4. Geometric Approach:**
* **Triangle Formation:** The bounded area is a triangle.
* **Base of the Triangle:** The base of the triangle is the distance between the two intersection points:
* Base = |(2 + a) / 3 - a| = |(2 - 2a) / 3| = (2 - 2a) / 3 (since a<2).
* **Height of the Triangle:**
* The height is the vertical distance from the vertex of f(x) (1, 1) to the line g(x).
* The x-coordinate of the vertex of f(x) is x=1.
* The y-coordinate of g(x) at x=1 is g(1)=2-a.
* The Height is 1-(2-a) = a-1
* **Area of the Triangle:**
* Area = (1/2) * Base * Height
* Area = (1/2) * [(2 - 2a) / 3] * (a - 1)
* Area = (1/6) * (2 - 2a) * (a - 1)
* Area = (-1/3) * (a - 1) * (a - 1)
* Area = (-1/3) * (a - 1)^2
* Area = (1/3) * (1-a)^2
**5. Prove Area < 1/3:**
* Since a ∈ (1, 2), (1 - a) is a negative value.
* Therefore (1-a)^2 is a positive value.
* Since a is between 1 and 2 then 0 < (a-1) < 1.
* Therefore 0 < (1-a)^2 < 1.
* Therefore 0 < (1/3)*(1-a)^2 < 1/3.
* Therefore the area is less than 1/3.
**Conclusion:**
For any value of 'a' in the interval (1, 2), the area bounded by the graphs of f(x) and g(x) is less than 1/3. This is proven using geometric properties of the graphs and without the use of differentiation or integration.
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