Problem on Step functions 2

Algebra Level 4

Find all real solutions of:

$\large \lfloor x\rfloor^{\lceil x \rceil} + \lceil x \rceil^{\lfloor x \rfloor} = \left(\lfloor x\rfloor + \lceil x \rceil\right)^{\lfloor x\rfloor \lceil x \rceil}$

If the solution set $S = (a,b]\cup \{c\}$ , find $a + b + c$ .

Notes:

Note that $0^0$ is undefined.
$\lfloor \cdot \rfloor$ denotes the floor function .
$\lceil \cdot \rceil$ denotes the ceiling function .

Also try Problem on Step functions .

The answer is 0.

1 solution

Dan Czinege
Apr 23, 2020

I got a=0, b=1, c=-1, is it correct?

That's what I got.

I don't see why the interval $(-1,0)$ is excluded. What is $(-1)^0$ ? That equals $1$ , right?

So $(-1)^0 + 0^{-1} = 1 = (-1 + 0)^{0*(-1)}$ , right? The LHS is $(-1)^{0}$ and the RHS is also $(-1)^{0})$ What am I missing?

Seems like solution set should be $[-1,0) \cup (0,1]$

Richard Desper - 1 year, 1 month ago

I think $0^{-1} = \Large\frac{1}{0}$ which is not defined. That's why I excluded this case.

Shikhar Srivastava - 1 year, 1 month ago

Then why do you include $0$ in $(-1,0]$ and exclude $-1$ ?

A Former Brilliant Member - 1 year, 1 month ago

@A Former Brilliant Member – I have not included 0. The solution set is $(0,1]\cup\{-1\}$ . So, $a = 0, b = 1, c = -1$

Shikhar Srivastava - 1 year, 1 month ago

Problem on Step functions 2

The answer is 0.

1 solution

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