Question

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. ex = 3 − 2x, (0, 1) The equation ex = 3 − 2x is equivalent to the equation f(x) = ex − 3 + 2x = 0. f(x) is continuous on the interval [0, 1], f(0) = _____, and f(1) = _____. Since f(0) < 0 < f(1) , 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 = 3 − 2x, in the interval (0, 1)

Answer 1

`f(0)=-2\\f(1)=e-1`

Explanation:

According to intermediate value theorem, if a function is continuous on an interval
`[a,b]`, and if
`k` is any number between
`f(a)` and
`f(b)`, then there exists a value,
`x=m`, where
`a<m<b`, such that
`f(m)=k`

In the given question,

Intermediate Value Theorem is used to show that there is a root of the given equation in the specified interval.

Here,

`f(x)=e^x-3+2x`

Put
`x=0`

`f(0)=e^0-3+2(0)=1-3+0=-2`

Put
`x=1`

`f(1)=e^1-3+2(1)=e-3+2=e-1`