Question

Find f(3) and f'(3), assuming that the tangent line to y = f(x) at a = 3 has equation

y = 6x + 8

Answer 1

The equation of the tangent line to a curve y = f(x) at x = a is given by y - f(3) = f'(a)(x - x1)
y = 6x + 8 . . . (1)
At x = 3, y = 6(3) + 8 = 18 + 8 = 26.

y - f(3) = f'(3)(x - 3)
y - f(3) = f'(3)x - 3f'(3)
y = f'(3)x - 3f'(3) + f(3) . . . (2)

Comparing (1) and (2), we have
f'(3) = 6, and
-3f'(3) + f(3) = -3(6) + f(3) = -18 + f(3) = 8
f(3) = 8 + 18 = 26

Therefore, f(3) = 26 and f'(3) = 6.