Question
If the tangent line to y = f(x) at (6, 5) passes through the point (0, 4), find f(6) and f '(6).
f(6) =
f '(6) =
f(6) =
f '(6) =
Answers
GPT-4o mini
To find \( f(6) \) and \( f'(6) \), let's analyze the problem step by step.
1. **Identifying \( f(6) \)**: Since the point given is \( (6, 5) \), it can be interpreted as the output of the function \( f \) at \( x = 6 \). Therefore,
\[
f(6) = 5.
\]
2. **Finding the slope of the tangent line**: The tangent line at the point \( (6, 5) \) passes through the point \( (0, 4) \). To find the slope of this line, we use the formula for the slope between two points:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 5}{0 - 6} = \frac{-1}{-6} = \frac{1}{6}.
\]
Thus, the slope of the tangent line at \( x = 6 \) is \( \frac{1}{6} \).
3. **Identifying \( f'(6) \)**: The derivative of the function at \( x = 6 \) gives us the slope of the tangent line at that point. Therefore,
\[
f'(6) = \frac{1}{6}.
\]
In conclusion, we have:
- \( f(6) = 5 \)
- \( f'(6) = \frac{1}{6} \)
So the final answers are:
\[
f(6) = 5,
\]
\[
f'(6) = \frac{1}{6}.
\]
1. **Identifying \( f(6) \)**: Since the point given is \( (6, 5) \), it can be interpreted as the output of the function \( f \) at \( x = 6 \). Therefore,
\[
f(6) = 5.
\]
2. **Finding the slope of the tangent line**: The tangent line at the point \( (6, 5) \) passes through the point \( (0, 4) \). To find the slope of this line, we use the formula for the slope between two points:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 5}{0 - 6} = \frac{-1}{-6} = \frac{1}{6}.
\]
Thus, the slope of the tangent line at \( x = 6 \) is \( \frac{1}{6} \).
3. **Identifying \( f'(6) \)**: The derivative of the function at \( x = 6 \) gives us the slope of the tangent line at that point. Therefore,
\[
f'(6) = \frac{1}{6}.
\]
In conclusion, we have:
- \( f(6) = 5 \)
- \( f'(6) = \frac{1}{6} \)
So the final answers are:
\[
f(6) = 5,
\]
\[
f'(6) = \frac{1}{6}.
\]