Question

Prove that |sinx|≤|x| using the mean value theorem

Answer #1

Use
the Mean Value Theorem to prove that -x < sin(x) < x, for x
> 0.

Use the Mean Value Theorem prove that sin x ≤ x for all x >
0

Use the Intermediate Value Theorem and the Mean Value Theorem to
prove that the equation cos (x) = -2x has exactly one real
root.

Prove the mean value theorem

Prove using Mean Value Theorem that if f' is bounded
then f is bounded too.

MATHS CALCULUS
I am asked to prove sinx <= x for all x>=0, by using the
derivative test for increasing and decreasing functions. I dont
know how to do this.

Use the Mean Value Theorem and the fact that for f(x) = cos(x),
f′(x) = −sin(x), to prove that, for x, y ∈ R,
| cos x − cos y| ≤ |x − y|.

Show that f(x) = x sin(x) has critical point on (0,pi)
by using mean value theorem.

prove that these functions are uniformly continuous on
(0,1):
1. f(x)=sinx/x
2. f(x)=x^2logx

Verify that f(x)= ln(x) satisfies the hypothesis of the Mean
Value Theorem on the interval [1,6] and find all numbers c that
satisfy the conclusion of the Mean Value Theorem.

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