AM-GM..... Huh?

Algebra Level 3

Find the minimum value of $4x^3+\dfrac{1}{x^{12}}$ where $x \in \mathbb{R_+}$ .

Bonus: Find the value of $x$ for which the expression has a minimum value.

1 solution

Rishabh Jain
Apr 2, 2016

Let expression be $\large\color{#20A900}{\mathfrak T}$ which can be written as:

$\large\underbrace{\color{#D61F06}{x^3+x^3+x^3+x^3}+\dfrac{1}{x^{12}}}_{\color{#EC7300}{\text{By AM }\geq \text{ GM}}}$

$\Large \color{#20A900}{\mathfrak T}\geq 5(1)^{1/5}=\color{#302B94}{\boxed{5}}$

Equality occurs when $x^3=\dfrac{1}{x^{12}}$ or $\color{#3D99F6}{x=1}$

Has there ever been a case when I post an algebra question and you don't write a solution to it? Anyway, upvoted!!

Rishik Jain - 5 years, 2 months ago

Yep, he's kinda solution master.

Aditya Sky - 5 years, 2 months ago

Rishabh Jain - 5 years, 2 months ago

Haha... Thanks... :-)

Rishabh Jain - 5 years, 2 months ago

I was about to post the same solution ! :)

Aditya Sky - 5 years, 2 months ago

Great .... :-) ..... Anyway I did your job.... :-)

Rishabh Jain - 5 years, 2 months ago