Key Concepts and Formulas

• Monotonicity and Roots: A strictly monotonic continuous function can intersect the x-axis at most once. If the range spans all real numbers, it must intersect exactly once.
• Derivative and Monotonicity: If $f'(x) > 0$ for all $x$, then $f(x)$ is strictly increasing. If $f'(x) < 0$ for all $x$, then $f(x)$ is strictly decreasing.
• Limits of Polynomials: The end behavior of a polynomial is determined by its highest-degree term. For odd-degree polynomials, $\lim_{x \to \infty} f(x) = \pm \infty$ and $\lim_{x \to -\infty} f(x) = \mp \infty$.

Step-by-Step Solution

Step 1: Define the function. Let the given equation be $f(x) = 0$. We define the function as: $f(x) = x^7 + 14x^5 + 16x^3 + 30x - 560$ Explanation: We define the left-hand side of the equation as a function $f(x)$ for easier analysis. The roots of $f(x) = 0$ correspond to the x-intercepts of the graph of $y = f(x)$.

Step 2: Calculate the first derivative of the function. To determine the monotonicity of $f(x)$, we calculate its first derivative, $f'(x)$. $f'(x) = \frac{d}{dx}(x^7 + 14x^5 + 16x^3 + 30x - 560)$ Applying the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$: $f'(x) = 7x^6 + 70x^4 + 48x^2 + 30$ Explanation: The derivative $f'(x)$ indicates the rate of change of $f(x)$. A positive derivative implies an increasing function, while a negative derivative implies a decreasing function.

Step 3: Analyze the sign of the derivative. We analyze the expression for $f'(x)$: $f'(x) = 7x^6 + 70x^4 + 48x^2 + 30$ Each term is non-negative:

• $x^6 \ge 0$, so $7x^6 \ge 0$.
• $x^4 \ge 0$, so $70x^4 \ge 0$.
• $x^2 \ge 0$, so $48x^2 \ge 0$. The constant term is positive: $30 > 0$. Therefore, $f'(x) > 0$ for all $x \in \mathbb{R}$. Explanation: Since each term involving $x$ has an even power, they are always non-negative. The addition of a positive constant ensures that the entire expression for $f'(x)$ is strictly positive.

Step 4: Determine the function's monotonicity. Since $f'(x) > 0$ for all $x \in \mathbb{R}$, the function $f(x)$ is strictly increasing over its entire domain $\mathbb{R}$. Explanation: A strictly positive derivative implies that the function is always increasing. As $x$ increases, $f(x)$ always increases.

Step 5: Evaluate the limits at positive and negative infinity. We examine the behavior of $f(x)$ as $x$ approaches $\infty$ and $-\infty$. $f(x) = x^7 + 14x^5 + 16x^3 + 30x - 560$ As $x \to \infty$: $\lim_{x \to \infty} f(x) = \lim_{x \to \infty} (x^7 + 14x^5 + 16x^3 + 30x - 560) = \infty$ As $x \to -\infty$: $\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} (x^7 + 14x^5 + 16x^3 + 30x - 560) = -\infty$ Explanation: The limits at infinity are determined by the term with the highest power, which is $x^7$. Since the power is odd, the limits at positive and negative infinity have opposite signs.

Step 6: Conclude the number of real solutions. We have established that:

1. $f(x)$ is strictly increasing on $\mathbb{R}$.
2. $\lim_{x \to -\infty} f(x) = -\infty$ and $\lim_{x \to \infty} f(x) = \infty$. Since $f(x)$ is a continuous function (a polynomial), and its range is $(-\infty, \infty)$, it must cross the x-axis exactly once. Thus, the equation $f(x) = 0$ has exactly one real solution.

Common Mistakes & Tips:

• Monotonicity is Key: Monotonicity arguments are powerful for determining the number of real roots, especially for higher-degree polynomials.
• Odd vs. Even Degree: Odd-degree polynomials always have at least one real root because their limits at $\pm \infty$ are $\pm \infty$ (or vice versa). Be cautious with even-degree polynomials, as they may have zero or multiple roots.
• Check Derivative Sign: Always explicitly check the sign of the derivative. Don't assume a polynomial is monotonic without verifying.

Summary

We found the number of real solutions to the equation $x^7 + 14x^5 + 16x^3 + 30x - 560 = 0$ by showing that the function $f(x) = x^7 + 14x^5 + 16x^3 + 30x - 560$ is strictly increasing and spans from $-\infty$ to $\infty$. Therefore, it has exactly one real root.

The final answer is $\boxed{1}$, which corresponds to option (B).