Question

Explain why the function has a zero in the given interval [1,2]
f(x)= (1/16) x^4−x^3+3
if f(1) is greater than 0 and f(2) is less than 0
I know it has to do with the intermediate values theorem. Is there anyone that can explain this theorem in plain terms?

Answer 1

Essentially, the IMV theorem says that for a continuous function on an interval [a, b] if
f(a)=y, f(b)=y', and y does not equal y', then there must be some c so that a is less than c, b is greater than c, and f(c) is in between y and y'.

So in your problem, use a=1, and b=2. Then, since f(1) is greater than 0, and f(2) is less than 0, 0 is in between f(1) and f(2). So by the IMV, there must be a c so that 1 is less than c is less than 2 such that f(c)=0. Therefore, f(x) has a root in the interval [1, 2].

Is it clearer now?

Answer 2

Yes thank you.