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The problem asks us to find the maximum area of a triangle formed in the first quadrant by the \( x \)-axis, \( y \)-axis, and a tangent line to the graph of \( f(x) = (x + 7)^{-2} \).

### Step-by-step approach:

1. **Find the derivative of \( f(x) \):**
   \[
   f(x) = (x + 7)^{-2}
   \]
   Using the power rule:
   \[
   f'(x) = -2(x + 7)^{-3}
   \]
   This gives the slope of the tangent line at any point on the curve.

2. **Equation of the tangent line:**
   The equation of the tangent line at a point \( (x_0, f(x_0)) \) on the curve is:
   \[
   y - f(x_0) = f'(x_0)(x - x_0)
   \]
   Substituting \( f(x_0) = (x_0 + 7)^{-2} \) and \( f'(x_0) = -2(x_0 + 7)^{-3} \):
   \[
   y - (x_0 + 7)^{-2} = -2(x_0 + 7)^{-3}(x - x_0)
   \]

3. **Find the intercepts:**
   To find where the tangent line intersects the \( x \)-axis and \( y \)-axis, we solve for the intercepts:
   
   - For the **x-intercept**: Set \( y = 0 \) in the tangent line equation.
   - For the **y-intercept**: Set \( x = 0 \) in the tangent line equation.

4. **Area of the triangle:**
   Once we have the \( x \)-intercept \( a \) and \( y \)-intercept \( b \), the area \( A \) of the triangle formed by these intercepts is:
   \[
   A = \frac{1}{2} \times a \times b
   \]

5. **Maximize the area:**
   We then maximize \( A \) as a function of \( x_0 \) by taking its derivative and solving for the critical points.

Let me calculate the area formula and the critical points next. Would you like further clarification on any of these steps so far?

Find the maximum area of a triangle formed in the first quadrant by the x-axis, y-axis, and a tangent line to the graph of f(x) = (x + 7)^{-2}.

Learn how to maximize the area of a triangle formed in the first quadrant by the x-axis, y-axis, and a tangent line to the curve f(x) = (x + 7)^{-2}. This problem involves calculus, derivatives, and optimization, suitable for advanced high school students.

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