MVT question + fixed points

[nugget]

Homework Statement

Consider the function g: [0,∞) -> R defined by G(x)=x+e^-2x

A. Use the mean value theorem to prove that |g(x₂)-g(x₁)|<|x₂-x₁| for all x₁,x₂ E [0,∞) with x₁≠x₂.

B. Find all fixed points of g on [0,∞).

Homework Equations

MVT: f'(c) = f(b)-f(a)/b-a.

The Attempt at a Solution

It seems we are being asked to prove in part A. that |g(x₂)-g(x₁)|/|x₂-x₁|<1.

This should be proved if we show that g'(x)<1 for x E [0,∞).

g'(x) = 1-2e^-2x. By inspection g'(x) is clearly less than 1 for x values between 0 and ∞.

Is this sufficient an explanation? M.V.T. states that the gradient of a function must, at at least one point (c,f(c)) between the points (a,f(a)) and (b,f(b)), equal the slope over the whole region [a,b] (as the equation shows). My method is essentially the same but replaces the 'rise over run' bit of the MVT simply with f'(x).

Am I on the right track, is there a better way to explain this, perhaps more directly involving MVT...? Thanks

have not yet tackled question B but any hints would be appreciated

 

[mighty2000]

.
Your explanation for part A using the MVT is correct. By showing that g'(x) is always less than 1 for all x in [0,∞), you have proven that the slope of the function is always less than the slope of the secant line connecting any two points on the function. This means that the function is always "flatter" than the secant line, which is why the absolute value of the difference between the function values is always less than the absolute value of the difference between the x values.

As for part B, a fixed point of a function is a value of x that, when plugged into the function, gives the same value back. In other words, g(x) = x. So to find the fixed points of g on [0,∞), you need to solve the equation x+e^-2x = x for x. This can be done algebraically by subtracting x from both sides and isolating the exponential term, or graphically by plotting the two functions y=x and y=x+e^-2x and finding their intersection points.

I hope this helps. Keep up the good work in your scientific studies!