State whether the following statements are true or false. Give a Short

Question

State whether the following statements are true or false. Give a Short proof or a counter-example in support of your answer.
a) The equation $x^3 - 4x - 16 = 0$ has not root in the interval [3, 5].
b) $\Delta E = E \Delta$ , where E is the shift operator and $\Delta$ is the forward difference operator.
c) Every $3 \times 3$ system of linear equations can be solved using the LU decomposition method.
d) For the data (2, 4), (1, 5), (3, 6) the Newton's divided difference f[x₀, x₁, x₂] is $\frac{3}{2}$ .
e) The Newton-Raphson method cannot be used to find a cube root of a positive real number.

09 Jan 2026

Answer :

Word Count : 210

a) True – If the function (f(x)) is continuous on [3, 5] and the given equation does not change sign in this interval, by Bolzano’s theorem (Intermediate Value Theorem), it has no root there. Counter-example: ________ ______ ______ __________ _____ ___ ___.
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