Question

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?

• A. Both statements (I) and (II) are FALSE.
• B. Both statements (I) and (II) are TRUE.
• C. Statement (I) is TRUE and statement (II) is FALSE.
• D. Statement (I) is FALSE and statement (II) is TRUE.

Hint:

Since all coefficients are positive and x is positive, the second derivative is positive, so the function is convex. Use this to judge the sign of the forward and backward difference errors.

Answer 1

The Correct Option is A

Step 1: Picture the curve near the point of interest.
Since every coefficient of this degree-5 polynomial is positive and $x>0$, the graph of $f$ bends upward (is convex) throughout this region; $f''(x)$ is a sum of positive terms for $x>0$, so $f''(x)>0$ everywhere here.

Step 2: Interpret the forward difference as a secant line's slope.
The forward difference $\dfrac{f(x+h)-f(x)}{h}$ is exactly the slope of the straight line (secant) joining the points $(x,f(x))$ and $(x+h,f(x+h))$ on the curve. For a convex curve, a secant drawn just ahead of a point always lies above the tangent line at that point, so its slope is steeper than the tangent's slope, meaning the forward difference OVER-estimates the true derivative, never under-estimates. This already makes statement (I) FALSE.

Step 3: Interpret the backward difference the same way.
The backward difference $\dfrac{f(x)-f(x-h)}{h}$ is the slope of the secant joining $(x-h,f(x-h))$ and $(x,f(x))$, the secant drawn just behind the point. For a convex curve this trailing secant is always less steep than the tangent at $x$, since the curve was rising more gently just before $x$ than it is rising exactly at $x$. So the backward difference UNDER-estimates the true derivative, never over-estimates. This makes statement (II) FALSE too.

Step 4: Put both pictures together.
For any convex function, the forward-difference secant always over-shoots the tangent slope and the backward-difference secant always under-shoots it, this follows from the mean value theorem applied on either side of the point. Both given statements describe the reverse behaviour, so both are wrong.

Final Answer:
Both statement (I) and statement (II) are FALSE, option (A). \[ \boxed{\text{Both (I) and (II) are FALSE}} \]