Question

If f(x) is a quadratic polynomial such that f(0) = 3, f'(2) = 2 and f'(3) = 12 then find f(x)

Answer 1

Let f(x) = ax² + bx + c;  a ≠ 0, a, b, c ∈ R

f'(x) = `"d"/("d"x) ("a"x^2 + "b"x + "c")` = a × 2x + b × 1 + 0 = 2ax + b

It is given that f(0) = 3, f'(2) = 2 and f'(3) = 12

But f(0) = a(0) + b(0) + c = c, f'(2) = 2a(2) + b = 4a + b and f'(3) = 2a(3) + b = 6a + b

∴ c = 3

4a + b = 2     ...(1)

and 6a + b = 12    ...(2)

Subtracting (1) from (2), we get 2a = 10

∴ a = 5

∴ from (1), 4(5) + b = 2

∴ b = 2 – 20 = – 18

∴ f(x) = 5x² – 18x + 3