Question

Solve the inequation : -8 1/2 <-1/2-4x《7 1/2; x€I​

Answer 1

Answer:

Proof: We break this proof into three cases.

Case 1: Suppose that x=0. Then clearly 0=sin(0)≤x=0.

Case 2: Suppose that 0<x<1. Let f(t)=sin(t). Then f is continuous and differentiable everywhere. In particular, f is continuous on [0,x] and differentiable on (0,x). By the Mean Value theorem there exists number c∈(0,1) such that:

(1)

f′(c)=f(x)−f(0)x−0

The derivative of sinx is cosx. Therefore:

(2)

cos(c)cos(c)=sin(x)−sin(0)x=sinxx

Note that the cosine function is bounded, that is,−1≤cost≤1 for every real number t. Therefore:

(3)

−1≤sinxx≤1⇒sinxx≤1

Since 0<x<1, we multiply both sides of the inequality above to get that sinx≤x.

Case 3: Suppose that 1≤x<∞. We know that sint is a bounded function and −1≤sint≤1 for every real number t. Thus sinx≤1≤x, i.e., sinx≤x. ■

Step-by-step explanation:

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