Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. ex = 9 − 8x, (0, 1)
The equation
ex = 9 − 8x
is equivalent to the equation
f(x) = ex − 9 + 8x = 0.

f(x)
is continuous on the interval
[0, 1],

f(0) =
,
and
f(1) =
.
Since

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< 0 <
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,
there is a number c in
(0, 1)
such that
f(c) = 0
by the Intermediate Value Theorem. Thus, there is a root of the equation
ex = 9 − 8x,
in the interval
(0, 1)

Which of the following are parts of a sphere

What is the first step of an indirect proof of a number that is divisible by two

1- is the product of two rational numbers irrational or rational? first, make a hypothesis by multiplying two rational numbers. then, use variables such as x=a/b and y=c/d and the closure property of integers to prove your hypothesis. 2- what do you think the product of a nonzero rational number and an irrational number is? is it rational or irrational? make use of variables, the closure property of integers, and possibly a proof by contradiction to prove your hypothesis. 3- why do we have to specify that the rational number must be nonzero when we determine what the product of a nonzero rational number and an irrational number is? if the rational number were 0, would it give us the same result we found in part b?