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

Given curve y = f(x) passes through (5, 4)

If any curve passes through a point then the equation of a curve must satisfy the point.

So f(5) = 4

Also tangent through (0, 3) touches the curve at (5, 4)

Slope of the tangent f^'(x) by using two-point form is m = f^'(x) = (y\(_2\) - y\(_1\) )/(x\(_2\) - x\(_1\) )

Given: x\(_1\) = 0, y\(_1\) = 3, x\(_2\) = 5, y\(_2\) = 4

So f^'(x) with these points gives

f^'(5) = (4 - 3)/(5 - 0) = 1/5

Thus f(5) = 5 and f'(x) = 1/5

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

Summary :

If the tangent line to y = f(x) at (5, 4) passes through the point (0, 3), then f(5) = 4 and f'(5) = 1/5.