Question

If the tangent line to y = f(x) at (5, 2) passes through the point (0, 1), find f(5) and f '(5).

Answer 1

f(5) = 2

f'(5) =
`(1)/(5)`

Explanation:

Tangent line to a function y = f(x) on a point (5, 2) passes through two points (5, 2) and (0, 1)

Let the equation of the line is,

y - y' = m(x - x')

Slope of a line passing through
`(x_1,y_1)` and
`(x_2,y_2)` =
`(y_2-y_1)/(x_2-x_1)`

=
`(2-1)/(5-0)`

=
`(1)/(5)`

Therefore, equation of the line passing through (0, 1) and slope =
`(1)/(5)` will be,

y - 1 =
`(1)/(5)(x-0)`

y =
`(x)/(5)+1`

Function representing equation will be,

f(x) =
`(x)/(5)+1`

At x = 5,

f(5) =
`(5)/(5)+1`

= 1 + 1

= 2

f(5) = 2

f'(x) =
`(d)/(dx)((x)/(5)+1)`

=
`(1)/(5)`

Therefore, f'(5) =
`(1)/(5)` will be the answer.