Question

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

Answer #1

for the equation f(x) = e^x - cos(x) + 2x - 3
Use Intermediate Value Theorem to show there is at least one
solution.
Then use Mean Value Theorem to show there is at MOST one
solution

Use the intermediate value theorem to prove that the
equation
ln? = ? − square root(?) has atleast one solution between ?=2
and ?=3

(i) Use the Intermediate Value Theorem to prove that there is a
number c such that 0 < c < 1 and cos (sqrt c) = e^c- 2.
(ii) Let f be any continuous function with domain [0; 1] such
that 0smaller than and equal to f(x) smaller than and equal to 1
for all x in the domain. Use the Intermediate Value Theorem to
explain why there must be a number c in [0; 1] such that f(c)
=c

x^5 +x^3 +x +1=0
use the IVT and Rolle's theorem to prove that the equation has
exactly one real solution.

Use the intermediate value theorem (only) to prove that any real
polynomial of odd degree has at least one real solution. Is the
same conclusion true if the degree is even?

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|.

use Rolle"s theorem to prove that 2x-2-cosx=0 has exactly one
real solution

1. Use the Intermediate Value Theorem to show that
f(x)=x3+4x2-10 has a real root in the
interval [1,2]. Then, preform two steps of Bisection method with
this interval to find P2.

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

Use intermediate theorem to show that theer is a root of
f(x)=-e^x+3-2x in the interval (0, 1)

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