Question

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.ex = 8 − 7x, (0, 1) The equation ex= 8 − 7x is equivalent to the equation f(x) = ex− 8 + 7x = 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= 8 − 7x, in the interval (0, 1).

Answer 1

Answer:

There is not root of the function $8-7x$ on the interval [0, 1].

Step-by-step explanation:

The Intermediate Value Theorem says:

If f(x) is a continuous function on [a, b], then for every d between f(a) and f(b), there exists a value c in between a and b such that f(c) = d.

This means that whenever we can show that:

• there is a point above some line
• and a point below that line, and
• that the curve is continuous,

we can then safely say yes, there is a value somewhere in between that is on the line.

To show that there is solution $8-7x=0$ between x = 0 and x = 1.

We evaluate the function at x = 0 and x = 1

$f(0)=8-7(0)=8\\f(1)=8-7(1)=1$

We know that

at x = 0, the line is above zero and

at x = 1 the line is above zero.

We also know that the function is continuous but because the line is always above zero there is no solution at the interval (0, 1).

We can check our work with the graph of the function.