Let's go through the problem and identify what needs to be corrected for part (c). The problem involves the function $f(x) = x^2 - x - 30$ and requests different tasks related to secant and tangent lines.

(a) Finding the equation of the secant line:

The secant line passes through the points $(-4, -10)$ and $(6, 0)$. The equation of the secant line is correctly calculated as:

$y = x - 6$

(b) Using the Mean Value Theorem:

The Mean Value Theorem guarantees a point $c$ in the interval $(-4, 6)$ such that the slope of the tangent line at $c$ is equal to the slope of the secant line. Since the slope of the secant line is $1$, we set the derivative of $f(x)$ equal to $1$.

$f'(x) = 2x - 1$

Setting $f'(c) = 1$:

$2c - 1 = 1 \implies c = 1$

(c) Finding the equation of the tangent line through $c$:

To find the tangent line at $c = 1$, we need to calculate the point on the curve and the slope of the tangent line at $x = 1$.

1. Find the slope of the tangent line at $x = 1$. We already know from part (b) that the slope at $c = 1$ is $1$.

2. Find the point on the curve at $x = 1$: $f(1) = 1^2 - 1 - 30 = -30$

So, the point on the curve is $(1, -30)$.

3. Write the equation of the tangent line. Using the point-slope form of a line: $y - y_1 = m(x - x_1)$ where $m = 1$, $x_1 = 1$, and $y_1 = -30$:

$y - (-30) = 1(x - 1)$ $y + 30 = x - 1$ $y = x - 31$

Conclusion:

The correct equation of the tangent line through $c = 1$ is:

$y = x - 31$

Would you like further details or have any questions about the process?

Here are 5 related questions to further your understanding:

1. How is the Mean Value Theorem used to find the point $c$?
2. Why is the slope of the secant line equal to the slope of the tangent line at $c$?
3. What is the geometric meaning of a secant line versus a tangent line?
4. How would you find the equation of the tangent line for a different function?
5. Can a function have more than one point where the Mean Value Theorem applies?

Tip: When finding the tangent line, always verify both the slope and the point on the curve to ensure the accuracy of your result.