Question

Show that the function `f(x)=4x^2-5x` has a zero between `1` and `2`.

Answer 1

As the value of function `f(x)` changes from negative to positive and as is continuous, it has a zero between `a` and `b`.

Explanation:

The function is `f(x)=4x^2-5x`.

Its derivative being `8x-5`, `f(x)` is a continuous function.

Further, at `a` i.e. `x=1`, the value of function is `4*1^2-5*1` or is `-1`.

At `b`, i.e. `x=2`, the value of function is `4*2^2-5*2` or is `6`.

It is obvious that between `a` and `b`, value of function changes from negative to positive and as it is continuous, it has a zero between `a` and `b`.

Answer 2

We may apply the Bolzano Theorem

Bolzano Theorem (BT)

Let, for two real a and b, a < b, a function f be continuous on a closed interval [a, b] such that f(a) and f(b) are of opposite signs. Then there exists a number `x_o`,in `[a, b]` with `f(x_o)=0`.

Hence we have that

`f(1)=4*1^2-5*1=-1`

`f(2)=4*2^2-5*2=6`

Because `f(1)*f(2)<0` then from Bolzano Theorem we have that there is a value `x_o` in `[1,2]` for which `f(x_o)=0`