True or False? (carefully stated)

Algebra Level 3

Let $x,y,z$ be real numbers satisfying $x+y+z=6$ and $xy+yz+zx=9$ .

Is it true that the maximum possible value of $(x-1) + (y-2)^2 + (z-3)^4$ is 88?

Yes No

1 solution

A Former Brilliant Member
May 23, 2019

Each of x, y and z lies in the range [0,4]. The combination (3,3,0) gives the maximum value of the given expression as 84. So it can not reach numbers greater than 84.

So the statement is always true, right? (I selected "yes" but was marked incorrect)

Chris Lewis - 2 years ago

No Chris. The expression can not take values like 85, 86, 87, 88. Or, any value in the range (84, 88].

A Former Brilliant Member - 2 years ago

Yes, I understand that. But the statement " $1\le 2$ " is true (you can even check it on Wolfram|Alpha).

The detail is that " $\le$ " is made up of two statements joined by a logical OR. You're right that it's never true that the expression is equal to $88$ . But it is always less than $88$ . So we have "(logical true) OR (logical false)" everywhere, which results in (logical true).

Put another way, when exactly is the statement in the question not true?

Chris Lewis - 2 years ago

Thanks. I've updated the problem statement for clarity. I've updated the answer to No . Those who previously answered Yes has been marked correct.

Brilliant Mathematics Staff - 2 years ago

True or False? (carefully stated)

1 solution

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