Understanding the Problem

We are given a polynomial f(x) of degree 4 with extreme values (local maxima or minima) at x = 1 and x = 2. This means its derivative f'(x) has roots at x = 1 and x = 2. Additionally, we are given the limit condition: $\underset{x\to 0}{\lim }[1+\frac{f\left(x\right)}{x^2}]$ = 3. We need to find the value of f(2).

Step-by-Step Solution

Step 1: General Form of the Polynomial

Since f(x) is a degree 4 polynomial, we can write it in its general form: $f\left(x\right)=a{x}^4+b{x}^3+c{x}^2+dx+e$, where a, b, c, d, e are constants and a ≠ 0.

Step 2: Using the Extreme Value Condition

Extreme values occur where the first derivative is zero. Let's find the derivative: $f^{\prime }\left(x\right)=4a{x}^3+3b{x}^2+2cx+d$.

Given that f'(1) = 0 and f'(2) = 0, we get two equations:

Equation 1 (from x=1): $4a(1{)}^3+3b(1{)}^2+2c\left(1\right)+d=0\Rightarrow 4a+3b+2c+d=0$

Equation 2 (from x=2): $4a\left(8\right)+3b\left(4\right)+2c\left(2\right)+d=0\Rightarrow 32a+12b+4c+d=0$

Step 3: Using the Limit Condition

The given limit is: $\underset{x\to 0}{\lim }[1+\frac{f\left(x\right)}{x^2}]=3$.

This simplifies to: $\underset{x\to 0}{\lim }\frac{f\left(x\right)}{x^2}=2$.

For this limit to exist and be finite (equal to 2), the numerator f(x) must approach 0 as fast as x² when x→0. This implies that the constant term and the coefficient of the x term in f(x) must be zero. Otherwise, the limit would be undefined (tending to infinity).

Therefore, we can deduce: $f\left(0\right)=0$ and the coefficient of x is also 0.

From the general form f(x) = ax⁴ + bx³ + cx² + dx + e, this gives us two more equations:

Equation 3 (constant term e): $e=0$

Equation 4 (coefficient d): $d=0$

Now, our polynomial simplifies to: $f\left(x\right)=a{x}^4+b{x}^3+c{x}^2$.

Let's substitute d=0 and e=0 into our derivative equations from Step 2:

Equation 1 becomes: $4a+3b+2c=0$ ...(1')

Equation 2 becomes: $32a+12b+4c=0$ ...(2')

We can simplify equation (2') by dividing by 4: $8a+3b+c=0$ ...(2'')

Step 4: Applying the Limit to Find Another Relation

We now use the simplified polynomial f(x)=ax⁴+bx³+cx² in our limit equation.

$\underset{x\to 0}{\lim }\frac{f\left(x\right)}{x^2}=\underset{x\to 0}{\lim }\frac{a{x}^4+b{x}^3+c{x}^2}{x^2}=\underset{x\to 0}{\lim }(a{x}^2+bx+c)$

Evaluating this limit as x approaches 0 gives us: $c=2$. This is our Equation 5.

Step 5: Solving the System of Equations

We now have three equations with three variables (a, b, c):

1. $4a+3b+2c=0$

2. $8a+3b+c=0$

3. $c=2$

Substitute c = 2 into equations 1 and 2:

Equation 1: $4a+3b+4=0\Rightarrow 4a+</$