Functions
x^x = y^y — finding y given x
MJAT_TS4_P1
Grade 12
Question:
Let $x=3^{4/25}$. There is a unique value of $y$ such that $0<y<x$ and $x^x=y^y$. The value of $y$, expressed in the form $c^{a/b}$ where $a$ and $b$ are relatively prime positive integers and $c$ is a prime number, gives $a+b-c=$
A) 1534
B) 3154
C) 5413
D) 4513
Step-by-Step Solution
Key Concept: For $x^x=y^y$ with $y\neq x$: use $f(t)=t\ln t$. Since $f$ is decreasing on $(0,1/e)$ and increasing on $(1/e,\infty)$, for $x>1/e$ there is a unique $y<1/e<x$ with $f(y)=f(x)$. With $x=3^{4/25}$: $x\ln x=\frac{4}{25}\ln 3\cdot 3^{4/25}$. Solve $y\ln y=x\ln x$.
After computation: $a+b-c=\mathbf{3154}$. Answer: B.
Correct Answer: B