Differentiability - primitive - integrals exercice

Differentiability - primitive - integrals

Exercice 1.— We have f : [a, b] → R a continuous function of [a, b], differentiable on ]a, b[ such as f (a) < 0 and f (b) > 0. We also assume that f verifies the following hypothesis:

(H) In every point x0∈]a, b[ as f (x0 ) = 0 we have f ′ (x0)> 0.

The object of the exercise is to show that the function f has a unique zero α ∈]a, b[.

1. Why are there at least one value α ∈] a, b [such that f (α) = 0?

We will now assume by contradiction (absurd) that there are two zeros a <α <β <b. The goal is to get a contradiction with the assumption (hypothese) (H).

2. Show that there exists γ ∈] α, β [such that f (γ)> 0.

We construct par recurrence two sequences (an) and (bn) in the following way. We set a0 = γ and b0 = β . We Suppose the suites we built to the rank n ≥ 0. We note c = (an +bn) / 2.

If f (c)> 0, then we set an a(n+1) = c and b(n+1) = bn . Otherwise, we set year a(n+1)=an and b(n+1)=c.

3. Verify that we construct two adjoining suites (suites adjacentes)

We note ω the common boundary of suites (an) and (bn).

4. Show that f (ω) = 0.

5. Using a limited development with the order 1, show that the fraction (f (bn)-f (an)) / (bn-an) tends to a limit that we will calculate. Deduce that f’(ω) ≤ 0 then conclude.

Application. We admit the existence of a function y ∈ C1 (R+) such that y(0) = -10 and checking for any t ≥ 0 the differential equation'

y '(t) = ln (1+ t^4+ t) - sin (y (t)^3).

We show that it has a unique z'ero on R+.

6. Check if there exists T> 0 such that y '(t) ≥ 1 for all t ≥ T. Deduce that there is at least one reel α> 0 such that y (α) = 0.

7. Using the above, conclude that y vanishes (s’annule) once and only one on R+