x^x+(7-x)^(7-x)=93

Question

What is x? Am thinking of using Lambert w function but am still lost on how to apply it

I Know that

W(xe^x)=x

Answer 1

This equation cannot be solved.

First derivative is:

x^x ∙ [ ln( x ) +1 ] + [ ln ( 7 - x ) -1 ] ∙ ( 7 - x )^( 7 - x )

First derivative = 0 in point x = 3.5

Second derivative is:

x^x ∙ [ ln ( x ) + 1 ]^2 + x^( x -1 ) - ( 7 - x )^( 6 - x ) - [ ln ( ( 7 - x ) + 1 ] ∙ ( ln ( 7 - x ) -1 ) ∙ ( 7 - x )^( 7 - x )

If second derivative > 0 in point where first derivative = 0 then function has minimum.

Second derivative in point x = 3.5 is:

38​8.49265 > 0

So the function has a minimum in point x​ = 3.5

y(min) = y(3.5) = 3.5 ^( 3.5) + ( 7 - 3.5 )^( 7 - 3.5 ) =

3.5 ^( 3.5) + 3.5 ^( 3.5) = 2 ∙ 3.5 ^( 3.5) =

2 ∙ 80.21178023 = 160.42356046

This function cannot have a value less than 160.42356046, so it cannot have a value of 93.

Answer 2

That derivative stuff gets kinda messy, but it should be clear that
f(x) = x^x + (h-x)^(h-x)
is symmetric about the line x = h/2.
Consider x=k such that k < h/2. Then
f(k) = k^k + (h-k)^(h-k)
and
f(h-k) = (h-k)^(h-k) + (h-(h - k))^(h-(h-k)) = (h-k)^(h-k) + k^k

Answer 3

Or
we could graph y = x^x and y = 93 - (7-x)^(7-x)
Wolfram shows them coming closest between 3 and 4, but
they never touch or intersect

https://www.wolframalpha.com/input/?i=plot+y+%3D+x%5Ex%2C+y+%3D+-%287-x%29%5E%287-x%29++%2B+93+from+3+to+4

Thus as Bosnian pointed out, there is no real solution