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

Answer:

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

Step-by-step 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 $a$, 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 $x=1$

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