To solve this problem, we can follow these steps:
1. Recognize that the tangent line to [tex]\( y = f(x) \)[/tex] at [tex]\((6, 5)\)[/tex] means that [tex]\( f(6) = 5 \)[/tex].
2. We need to find the slope of the tangent line, [tex]\( f'(x) \)[/tex], at [tex]\( x = 6 \)[/tex]. This slope can be determined by the points [tex]\( (6, 5) \)[/tex] and [tex]\( (0, 4) \)[/tex], which lie on the tangent line.
3. The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is calculated by the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
4. Plugging in the given points [tex]\((6, 5)\)[/tex] and [tex]\((0, 4)\)[/tex] into the formula to find the slope:
[tex]\[
m = \frac{4 - 5}{0 - 6} = \frac{-1}{-6} = \frac{1}{6}
\][/tex]
5. Thus, the slope of the tangent line at [tex]\( x = 6 \)[/tex], which is [tex]\( f'(6) \)[/tex], is [tex]\( \frac{1}{6} \)[/tex].
6. Therefore, the values are:
[tex]\[
f(6) = 5 \quad \text{and} \quad f'(6) = \frac{1}{6}
\][/tex]
The correct answer is:
c. [tex]\( f(6) = 5 \quad \text{and} \quad f'(6) = \frac{1}{6} \)[/tex]
