Question

How do you use the Intermediate Value Theorem to show that the polynomial function `f(x) = 10x^4 - 2x^2 + 7x - 1` has a root in the interval [-3, 0]?

Answer 1

See the explanation.

Explanation:

The Intermediate Value Theorem says that

if `f` is continuous on `[a,b]`, then `f` attains every value between `f(a)` and `f(b)` at `x` values in `[a,b]`

In general, to show that a function attains a chosen value, call it `N`, on interval `[a,b}` using the IVT requires:

Show that the function is continuous on `[a,b]`

Show that `N` is between `f(a)` and `f(b)`

Conclude, using the Intermediate Value Theorem, that there is a `c` in `[a,b]` with `f(c) = N`

In this case, `f(x)` has a root `c` if and only if `f(c) = 0`, so we need to show that there is a `c` in `[-3,0]` where `f(c) = 0`

`f` is continuous on `[-3,0}` because it is a polynomial and polynomials are continuous at every real number.

`0` is between `f(-3) and`f(0)# because #f(-3) = 10(81)-2(9)+7(-3)-1`is positive and`f(0) = -1# is negative.

Therefore, by the Intermediate Value Theorem, there is a `c` in `[-3,0]` with `f(c) = 0`.

This `c` is a root of `f(x)`.

Final Note In mathematics "there is a" mean "there is at least one". It does NOT mean "there is exactly one".
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