Question:

# Solution: Number of Real Roots of $e^x = x^n$ We analyze the equation $e^x = x^n$ for positive integers $n$ by studying the function: $$f(x) = e^x - x^n$$ The number of real roots equals the number of zeros of $f(x)$. We examine the behavior for different values of $n$. ## General Strategy **Critical observations:** - For $x \leq 0$: We have $e^x > 0$ always, while $x^n \leq 0$ when $n$ is odd and $x < 0$, or $x^n > 0$ when $n$ is even. - For $x > 0$: Both $e^x$ and $x^n$ are positive, requiring careful analysis. --- ## Case A: $n = 1$ We solve $e^x = x$. $$f(x) = e^x - x$$ $$f'(x) = e^x - 1$$ **Analysis:** - $f'(x) = 0$ when $e^x = 1$, giving $x = 0$ - $f'(x) < 0$ for $x < 0$ and $f'(x) > 0$ for $x > 0$ - Thus $f$ has a global minimum at $x = 0$ **Evaluating at the minimum:** $$f(0) = e^0 - 0 = 1 > 0$$ Since $f(x) \geq f(0) = 1 > 0$ for all $x \in \mathbb{R}$, we have $e^x > x$ for all real $x$. **Conclusion:** $\boxed{\text{0 real roots}}$ → **[A-s]** --- ## Case B: $n = 2$ We solve $e^x = x^2$. $$f(x) = e^x - x^2$$ $$f'(x) = e^x - 2x$$ **Behavior as $x \to -\infty$:** $$e^x \to 0 \text{ and } x^2 \to +\infty \implies f(x) \to -\infty$$ **Behavior as $x \to +\infty$:** $$e^x \text{ grows faster than } x^2 \implies f(x) \to +\infty$$ **Critical points:** Setting $f'(x) = 0$: $$e^x = 2x$$ Let $g(x) = e^x - 2x$. Then $g'(x) = e^x - 2$. - $g'(x) = 0$ when $x = \ln 2 \approx 0.693$ - $g(\ln 2) = 2 - 2\ln 2 \approx 2 - 1.386 = 0.614 > 0$ So $f'(x) = e^x - 2x > 0$ for all $x$ (minimum value is positive). **Verification of monotonicity:** Since $f'(x) \geq 0.614 > 0$ everywhere, $f$ is strictly increasing. A strictly increasing function with $f(-\infty) = -\infty$ and $f(+\infty) = +\infty$ crosses the $x$-axis exactly once. **Conclusion:** $\boxed{\text{1 real root}}$ → **[B-r]** --- ## Case C: Odd $n \geq 3$ We solve $e^x = x^n$ for odd $n \geq 3$. $$f(x) = e^x - x^n$$ **For $x < 0$:** - $e^x > 0$ always - $x^n < 0$ (since $n$ is odd and $x < 0$) - Therefore $f(x) = e^x - x^n > 0$ for all $x < 0$ So there are <b>no roots