Question

(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

Answer #2

answered by: anonymous

a fixed point of a function f is a number c in its domain such
that f(c)=c. use the intermediate value theorem to prove that any
continious function with domain [0,1] and range a subset of [0,1]
must have a fixed point.[hint: consider the function f(x)-x]
“Recall the intermediate value theorem:suppose that f is
countinous function with domain[a,b]and let N be any number between
f(a)and f(b), where f(a)not equal to f(b). Then there exist at
least one number c in...

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

1.- Prove the intermediate value theorem: let (X, τ) be a
connected topological space, f: X - → Y a continuous transformation
and x1, x2 ∈ X with a1 = f (x1), a2 = f (x2) ( a1 different a2).
Then for all c∈ (a1, a2) there is x∈ such that f (x) = c.
2.- Let f: X - → Y be a continuous and suprajective
transformation. Show that if X is connected, then Y too.

Prove that f(x)=x*cos(1/x) is continuous at x=0.
please give detailed proof. i guess we can use squeeze
theorem.

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

Prove the following theorem:
Theorem. Let a ∈ R
and let f be a function defined on an
interval centred at a.
IF f is continuous at a
and f(a) > 0 THEN
f is strictly positive on some interval
centred at a.

1. Does the function satisfy the hypotheses of the Mean Value
Theorem on the given interval?
f(x) = x3 + x − 5, [0, 2]
a) No, f is continuous on [0, 2] but not differentiable
on (0, 2).
b) Yes, it does not matter if f is continuous or
differentiable; every function satisfies the Mean Value
Theorem.
c) There is not enough information to verify if this function
satisfies the Mean Value Theorem.
d) Yes, f is continuous on [0,...

Find the value of C > 0 such that the function
?C sin2x, if0≤x≤π,
f(x) =
0, otherwise
is a probability density function.
Hint: Remember that sin2 x = 12 (1 − cos 2x).
2. Suppose that a continuous random variable X has probability
density function given by the above f(x), where C > 0 is the
value you computed in the previous exercise. Compute E(X).
Hint: Use integration by parts!
3. Compute E(cos(X)).
Hint: Use integration by substitution!

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

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?

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