Question

The first, second and third derivatives of a cubic polynomial f(x) at x = 1, are 1^3, 2^3 and 3^3 respectively. Then the value of f(0) + f(1) – 2f(-1) is?

(a) 76

(b) 86

(c) 126

(d) 41.5

This question was addressed to me in an international level competition.

My query is from The nth Derivative of Some Elementary Functions topic in division Differential Calculus of Engineering Mathematics

Answer 1

The correct answer is (d) 41.5

The explanation is: Assume the polynomial to be of the form f(x) = ax^3 + bx^2 + cx + d

Now the first derivative at x = 1 yields the following equation

1^3 = 1 = 3a + 2b + c

The second derivative at x = 1 yields the following expression

2^3 = 8 = 6a + 2b

The third derivative at x = 1 yields the following equation

3^3 = 27 = 6a

Solving for a, b and c simultaneously yields

(a, b, c) = (^9⁄2, ^-19⁄2, ^13⁄2)

Hence the assumed polynomial is f(x) = 9x^3 – 19x^2 + 13x ⁄ 2 + d

Now the given expression can be evaluated as

f(0) + f(1) – 2f(-1) = (d) + (^3⁄2 + d) – 2(-20 + d)

= 40 + ^3⁄2

= 41.5.