use the intermediate value theorem to show that the function has a zero in the interval (1.4, 1.5) f(x)= x^5-x^4+7x^3-7x^2-18x+18

Answer 1

f(x)= x^5-x^4+7x^3-7x^2-18x+18; f is a polynomial and so is continuous on (-inf,inf)

Evaluate f at 1.4: f(1.4) = -0.17536

Evaluate f at 1.5: f(1.5) = 1.40625

The I.V.T. states that If f is a function which is continuous on the closed interval [a,b] then there exists a value c in (a, b) such that f(a) < f(c) < f(b).

In this case f is continuous on [1.4,1.5] since it is continuous everywhere. So in non mathematical terms, if the function, a line, has no holes or gaps, and it must pass through each value between any two values where it is calculated. In our case, it was negative, somewhere (~ -.18) and it was positive somewhere (

~ 1.4 ) so it must be 1.3, -0.1, 1, and yes, even 0 somewhere!

So there you have it, there must be a zero on the interval. There might even be more than one. The theorem does not say that the phenomena described doesn't happen repeatedly, just that it happens.