If the tangent line to y = f(x) at (5, 4) passes through the point (0, 3), find f(5) and f'(5)

To solve this problem, we first need to understand the information provided about the tangent line. We know that the tangent line to the curve y = f(x) at the point (5, 4) can be expressed using the point-slope form of a line.

1. **Identifying the Point on the Tangent Line**: The point (5, 4) implies that f(5) = 4. This means that the function’s value at x = 5 is 4, so we can directly write:

f(5) = 4

2. **Finding the Slope of the Tangent Line**: The line also passes through the point (0, 3). We can use these two points, (5, 4) and (0, 3), to find the slope (m) of the tangent line.

The formula for the slope between two points (x₁, y₁) and (x₂, y₂) is:

m = (y₂ – y₁) / (x₂ – x₁)

Using our points (5, 4) and (0, 3):

m = (3 – 4) / (0 – 5) = (-1) / (-5) = 1/5

Thus, the slope of the tangent line at the point (5, 4) is 1/5, which means:

f'(5) = 1/5

In conclusion, the values we found are:

• f(5) = 4
• f'(5) = 1/5