x^x+(7-x)^(7-x)=93
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
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:
388.49265 > 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.
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
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
To solve the equation x^x + (7-x)^(7-x) = 93 using the Lambert W function, we first need to isolate the x term on one side of the equation.
Start by rearranging the equation:
x^x = 93 - (7-x)^(7-x)
Next, we want to eliminate the exponentiation on x^x. To do this, we take the natural logarithm of both sides of the equation:
ln(x^x) = ln(93 - (7-x)^(7-x))
The property ln(a^b) = b * ln(a) allows us to simplify the left side of the equation:
x * ln(x) = ln(93 - (7-x)^(7-x))
Now, we can use the Lambert W function to further solve this equation. Lambert W function is defined as follows: if z = W(x), then x = z * e^z.
Apply Lambert W to the equation:
x * ln(x) = ln(93 - (7-x)^(7-x))
Multiply both sides by (1/ln(x)):
x = (1/ln(x)) * ln(93 - (7-x)^(7-x))
Now we have an expression in the form x = (1/ln(x)) * f(x), where f(x) represents the term (93 - (7-x)^(7-x)).
To approximate the value of x, we can use an iterative method since there is no algebraic solution for this equation. Here's an outline of the iterative process:
1. Make an initial guess for x, for example, x = 1.
2. Evaluate f(x) based on the current value of x.
3. Calculate the next approximation for x using the formula:
x_next = (1/ln(x)) * f(x)
4. Repeat steps 2-3 until the value of x stabilizes (the difference between x and x_next is very small).
Using the iterative method to solve equations involving Lambert W can be complex and time-consuming. It may be more efficient to use numerical methods or computer algorithms to find an approximate solution.