Let f(x) be a polynomial of degree four, having extreme values at x = 1 and x = 2. If $l im_{x \to 0} [1 + \frac{f ( x )}{x ^{2}}] = 3$ , then f(2) is

Answer (Detailed Solution Below)View answer ▾

Answer: Option A — 0

Detailed Solution

The limit gives $\frac{f ( x )}{x ^{2}} \to 2$ , so f has no constant or x term and its $x^{2}$ coefficient is 2: $f (x) = a x^{4} + b x^{3} + 2 x^{2}$ .

$f' (x) = 4 a x^{3} + 3 b x^{2} + 4 x$ ; $f' (1) = 4 a + 3 b + 4 = 0$ and $f' (2) = 32 a + 12 b + 8 = 0$ .

Solving, $a = \frac{1}{2}$ and $b = - 2$ , so $f (x) = \frac{x ^{4}}{2} - 2 x^{3} + 2 x^{2}$ and $f (2) = 8 - 16 + 8 = 0$ .

✓ Check

f (x) = \frac{1}{2} x^{2} (x - 2)^{2}

, which is visibly 0 at

x = 2

, and

f' (x) = 2 x (x - 1) (x - 2)

vanishes at 1 and 2.

⚠ Common trap

Reading the limit as

\frac{f ( x )}{x ^{2}} \to 3

(forgetting the '1 +') changes only the scale of f; f(2) is still 0, so (B), (C), (D) cannot arise from a consistent working.

Quick tip

$f^{'} = k x (x - 1) (x - 2) \Rightarrow f = \frac{k}{4} x^{2} (x - 2)^{2}, f (2) = 0$

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Limits — Preparation

This question tests your understanding of Limits, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

The limit of a function describes the value the function approaches as its input approaches a given point. Limits are the foundation of calculus, because continuity, derivatives and integrals are all defined using them. This topic covers left-hand and right-hand limits, the algebra of limits, and standard methods of evaluation: direct substitution, factorisation, rationalisation, and dividing by the highest power for limits at infinity. It also covers the standard limits, such as (sin x)/x -> 1 as x -> 0, (e^x - 1)/x -> 1, (a^x - 1)/x -> log a, and (1 + x)^(1/x) -> e. Most exam questions are 0/0 forms that reduce to one of these after simplification. A limit exists only when the left-hand and right-hand limits are equal, which is especially important for the modulus and greatest integer functions. Practise the limits questions below, each with a step-by-step solution.

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Let f(x) be a polynomial of degree four, having extreme values at x = 1 and x = 2. If $l im_{x \to 0} [1 + \frac{f ( x )}{x ^{2}}] = 3$ , then f(2) is