for any real number

Algebra Level 2

Find the minimum value of $\dfrac{x^4 + 3}x$ , where $x$ is a positive real number.

The answer is 4.

1 solution

A Former Brilliant Member
Jun 1, 2020

The A. M. - G. M. inequality holds for positive reals only. So assuming that $x$ is positive, we proceed as follows :

The given expression is $\frac{x^4+3}{x}=x^3+\frac{1}{x}+\frac{1}{x}+\frac{1}{x}\geq 4\sqrt[4] {x^3\times \frac{1}{x}\times \frac{1}{x}\times \frac{1}{x}}=\boxed 4$ .

If we allow for any real x (including x<0), there should be no minimum. As x->0 from the left, (x^4 + 3)/x -> - infinity. I'm not sure what I am missing.

Ron Gallagher - 1 year ago

You are right, the domain should be restricted to positive reals only.

Elijah L - 1 year ago

Definitely. You aren't missing anything. The domain of $x$ must be positive reals.

A Former Brilliant Member - 1 year ago

for any real number

The answer is 4.

1 solution

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