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There is a french exercice which I translated in english. You might need to solve it by looking at the original to not skip anything, but if there is something unclear, please reffer to the traduction.

Please include ALL the calculs, demonstrations and the theories you reffer to, also I hope that you can guarantee 99%-100% correct and full answers.

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**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

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+