Question

How do you find all values of x such that f(x) = 0 given `f(x) = (1/4)x^3 - 2`?

Answer 1

Assuming `f:RR->RR`, then `x=2` is the only value which satisfies `f(x)=0`.

Explanation:

We wish to find the roots of the function

`f(x) = (1/4)x^3-2`

By root, we mean any value of `x` for which `f(x)=0`.

`:. (1/4)x^3-2=0 =>(1/4)x^3=2`

Multiply both sides by `4`.

`x^3=8=>color(red)(x=2)`

Let this first root we found be `x_1`. However, we did it algebraically and there seems to be only one solution. Could this be the only one?

Yes. If the function has domain and range over the real numbers `RR` then `x=2` is the only root of `f`.

We can verify this graphically, too:

graph{(y-1/4x^3+2)=0 [-7.9, 7.9, -3.31, 4.59]}