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Algebra Level 3

x 256 − 16 x 16 + 16 \large x^{256} - 16x^{16} + 16

What is the minimum value of the expression above? Can you generalize it?

This is part of the series: " It's easy, believe me! "


The answer is 1.

Thành Đạt Lê by Thành Đạt Lê , 17, Singapore

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1 solution

Let x 16 = y x^{16} = y

Using this substitution, the above expression can be wriitten as \text{Using this substitution, the above expression can be wriitten as} :

y 16 − 16 y + 16 y^{16} - 16y + 16

Differentiating the above expression we get \text{Differentiating the above expression we get} :

16 y 5 − 16 16y^{5} - 16

Clearly , the function is increasing for y>1 and decreasing for y<1. Hence the minima occurs at y = 1 or better to say , x = 1 \text{Clearly , the function is increasing for y>1 and decreasing for y<1. Hence the minima occurs at y = 1 or better to say , x = 1}

Putting the value , we get the minimum value of the expression as \text{Putting the value , we get the minimum value of the expression as} ,

1 − 16 + 16 = 1 1 - 16 + 16 = 1

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