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

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Answer 1 · 2018-06-01T11:22:57

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

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]}