Key Concepts and Formulas

• Inverse Functions: If $g(f(x)) = x$ for all $x$, then $g(x)$ is the inverse of $f(x)$, denoted as $g(x) = f^{-1}(x)$. This means if $f(a) = b$, then $g(b) = a$.
• Derivative of an Inverse Function: If $g(x)$ is the inverse of $f(x)$, then $g'(y) = \frac{1}{f'(x)}$, where $y = f(x)$.
• Monotonicity: If $f'(x) > 0$ for all $x$, then $f(x)$ is strictly increasing and has a unique inverse.

Step-by-Step Solution

Step 1: Determine the value of $g(7)$

• Goal: Find the value of $g(7)$ using the definition of the inverse function. We need to find $x$ such that $f(x) = 7$.
• Action: Set $f(x) = 7$ and solve for $x$: $x^5 + 2x^3 + 3x + 1 = 7$ $x^5 + 2x^3 + 3x - 6 = 0$
• Reasoning: We look for integer roots by testing factors of -6. Trying $x=1$: $(1)^5 + 2(1)^3 + 3(1) - 6 = 1 + 2 + 3 - 6 = 0$ So, $x=1$ is a root, and $f(1) = 7$. To confirm this is the unique solution, we check the monotonicity of $f(x)$.
• Uniqueness Check: Find the derivative of $f(x)$: $f'(x) = \frac{d}{dx}(x^5 + 2x^3 + 3x + 1) = 5x^4 + 6x^2 + 3$ Since $x^4 \ge 0$ and $x^2 \ge 0$ for all real $x$, we have $5x^4 \ge 0$ and $6x^2 \ge 0$. Thus, $f'(x) = 5x^4 + 6x^2 + 3 \ge 3 > 0$ Since $f'(x) > 0$ for all $x$, $f(x)$ is strictly increasing and has a unique inverse. Therefore, $x=1$ is the only solution to $f(x) = 7$.
• Conclusion: Since $f(1) = 7$, we have $g(7) = 1$.

Step 2: Determine the value of $g'(7)$

• Goal: Find the value of $g'(7)$ using the derivative of the inverse function formula.
• Action: Use the formula $g'(y) = \frac{1}{f'(x)}$, where $y = f(x)$. We want $g'(7)$, and we know $f(1) = 7$, so $y = 7$ and $x = 1$. Thus, $g'(7) = \frac{1}{f'(1)}$.
• Reasoning: We need to evaluate $f'(x)$ at $x=1$. We already calculated $f'(x)$ in Step 1: $f'(x) = 5x^4 + 6x^2 + 3$ Substitute $x=1$ into $f'(x)$: $f'(1) = 5(1)^4 + 6(1)^2 + 3 = 5 + 6 + 3 = 14$
• Conclusion: Therefore, $g'(7) = \frac{1}{f'(1)} = \frac{1}{14}$.

Step 3: Calculate $\frac{g(7)}{g'(7)}$

• Goal: Substitute the values of $g(7)$ and $g'(7)$ into the expression $\frac{g(7)}{g'(7)}$.
• Action: $\frac{g(7)}{g'(7)} = \frac{1}{\frac{1}{14}}$
• Reasoning: Dividing by a fraction is equivalent to multiplying by its reciprocal.
• Conclusion: $\frac{1}{\frac{1}{14}} = 1 \times 14 = 14$

Common Mistakes & Tips

• Inverse Function Definition: Remember that $g(f(x))=x$ implies $g$ is the inverse of $f$. This means if $f(a)=b$, then $g(b)=a$.
• Finding the Correct x: When calculating $g'(y)$, find $x$ such that $f(x) = y$ first, then use that $x$ in $f'(x)$ to find $g'(y) = 1/f'(x)$.
• Monotonicity Check: Verify that $f(x)$ is strictly increasing or decreasing to ensure there is a unique inverse and a unique $x$ such that $f(x) = y$.

Summary

We identified that $g(x)$ is the inverse of $f(x)$. We found $g(7)$ by solving $f(x)=7$ to get $x=1$, so $g(7)=1$. We then found $f'(x)$ and calculated $f'(1)=14$, which allowed us to calculate $g'(7) = 1/f'(1) = 1/14$. Finally, we calculated $\frac{g(7)}{g'(7)} = \frac{1}{1/14} = 14$.

Final Answer

The final answer is \boxed{14}, which corresponds to option (D).