The Hint is: (Use Algebra!!!)

Algebra Level 2

Find the minimum value of the given expression, if $a>0$

$a^{-5} + a^{-4} + 3a^{-3} + 1 + a^{8} + a^{10}$

The answer is 8.

1 solution

Raj Magesh
Aug 29, 2014

$S = a^{-5}+a^{-4}+a^{-3}+a^{-3}+a^{-3}+1+a^8+a^{10}$

Applying AM-GM, since $a>0$ ,

$\dfrac{S}{8} \ge \sqrt{a^{-5}\cdot a^{-4} \cdot a^{-3} \cdot a^{-3} \cdot a^{-3} \cdot 1 \cdot a^8 \cdot a^{10}}$

$S \ge 8$

$S_{min} = \boxed{8}$

Thanks! What I tried first was to differentiate the expression with respect to a and equate it to 0, trying to solve for a. But the only easily observed root is a=1, which isn't in our domain.

Another approach: split the domain into 2 parts, one from $(-1,0)$ and another from $(-\infty, -1]$ .

In the first part, if a approaches 0 from the left, $a^8$ and $a^{10}$ will become very small positive numbers, $a^{-4}$ will be a large positive number, and $a^{-5} +3a^{-3}$ will become a very large negative number (in magnitude), making the expression very negative.

In the second part, as a approaches negative infinity, $a^8$ and $a^{10}$ will become very large positive numbers, while the other terms tend to 0, making the expression very large and positive.

Hence, the maximum value should be positive infinity.The graph of the function corroborates this too.

Is there a better method?

Raj Magesh - 6 years, 9 months ago

Very good solution....and the easiest one too..

Kislay Raj - 6 years, 9 months ago

Excellent, did the same :)

Jayakumar Krishnan - 6 years, 9 months ago

I applied the AM-GM inequallity with 3a^(-3) as a term, what is the problem with this aproach? I found minima at a=1 and Smin=7.2 (aprox)

Carlos David Nexans - 6 years, 9 months ago

If you do that, you end up with:

$\dfrac{S}{6} \ge \left(3a^6\right)^{\dfrac{1}{6}}$

$S_{min} = 6a\sqrt[6]{3}$

This doesn't help us, since the minimum is in terms of a. The actual minimum is at a = 1.1102, according to WolframAlpha, making the minimum value of the expression 8.

So the trick is to make the stuff under the root independent of a by splitting the given terms. This question makes that pretty obvious (since only one conspicuous term has a coefficient of 3) but others are more subtle...

Raj Magesh - 6 years, 9 months ago

Great solution Raj. How would one find the maximum value of this expression analytically when a<0?

Edwin Hughes - 6 years, 9 months ago

The Hint is: (Use Algebra!!!)

The answer is 8.

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