If the tangent line to y = f(x) at (7, 6) passes through the point (0, 5), find f(7) and f'(7).

Solution:

Given y = f(x) ; two points (7,6) and (0,5)

We have equation of line passing through two points (y₂ - y₁) / (y- y₁) = (x₂ - x₁) / (x- x₁) (two-point form)

⇒ (5-6)/(y-6)= (0-7)/(x-7)

⇒ (-1)(x-7) = (-7)(y-6)

⇒ x-7 = 7y -42

⇒ 7y = x+35

⇒ y= (x/7) +5 [y =mx +b]

Slope = m = f'(x) = 1/7

Since f(x) =  (x/7) +5, we have

f(7) = 1+5 =6 ; f'(7) = 1/7

If the tangent line to y = f(x) at (7, 6) passes through the point (0, 5), find f(7) and f'(7).

Summary:

If the tangent line to y = f(x) at (7, 6) passes through the point (0, 5), then f(7) and f'(7) are 6 and 1/7 respectively.