Let f(x) be a polynomial of degree four having extreme values at x = 1 and x = 2. If = 3, then f(2) is equal to

Engineering

Mathematics

Methods to Evaluate Limits

Question

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

– 4

– 8

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We have $\underset{x \to 0}{Lim} \frac{f (x)}{x^{2}} = 2$

∴ f (x) = (ax⁴ + bx³ + 2x²)

Now, f ' (1) = 0 ⇒ 4a + 3b = – 4

and f' (2) = 0 ⇒ 8a + 3b = – 2

∴ we get, a = $\frac{1}{2}$ , b = – 2

So, f (2) = 0