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GATE 2026 CE (CE2) – Question 12

Engineering Mathematics · Numerical Methods: Error analysis, algebraic equations, interpolation, differentiation, integration, ODEs · 1 mark · Multiple choice

A fifth-degree polynomial in $x$ is defined for $x > 0$. All coefficients of the polynomial are positive. The first derivative of the polynomial is obtained numerically at a point by using the first-order forward as well as the first-order backward difference methods. Identical step lengths are used for both the methods. Following statements are made.

(I) Forward difference method underestimates the true derivative.

(II) Backward difference method overestimates the true derivative.

Which one of the following options is CORRECT?

  1. Both statements (I) and (II) are FALSE.
  2. Both statements (I) and (II) are TRUE.
  3. Statement (I) is TRUE and statement (II) is FALSE.
  4. Statement (I) is FALSE and statement (II) is TRUE.

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Correct answer: (A) Both statements (I) and (II) are FALSE.

Explanation

With all coefficients positive, $f''(x) = 20a_5x^3 + 12a_4x^2 + 6a_3x + 2a_2 > 0$ for $x > 0$, so the polynomial is convex. For a convex function the chord slope to the right of the point is larger than $f'(x)$ and the chord slope to the left is smaller. The forward difference therefore overestimates and the backward difference underestimates the derivative. Statement (I) says the opposite for the forward difference and statement (II) says the opposite for the backward difference, so both are false.