Question

Let \( f(x) \) be a polynomial of degree \( 5 \) having extreme values at \( x = 1 \) and \( x = -1 \). If \[ \lim_{x \to 0} \frac{f(x)}{x^3} = -5, \] then the value of \( f(B) - f(-2) \) is

• A. 110
• B. 112
• C. 115
• D. 118

Hint:

When a limit of the form \( \lim_{x\to 0}\frac{f(x)}{x^3} \) is finite, the polynomial must not contain terms of degree less than \( 3 \). Then use the derivative condition for maxima or minima.

Answer 1

The Correct Option is B

Step 1: Use the given limit condition.
We are given that \[ \lim_{x \to 0} \frac{f(x)}{x^3} = -5. \] This implies that near \( x = 0 \), the polynomial \( f(x) \) must contain \( x^3 \) as a factor, and the coefficient of \( x^3 \) must be \( -5 \). Since \( f(x) \) is a polynomial of degree 5, we can express it in the general form: \[ f(x) = ax^5 + bx^4 - 5x^3. \] There are no constant, linear, or quadratic terms; otherwise, the given limit would not exist as a finite value equal to \( -5 \).
Step 2: Apply the condition for extreme values at \( x = 1 \) and \( x = -1 \).
If the function has extreme values at \( x = 1 \) and \( x = -1 \), then the first derivative must be zero at those points: \[ f'(1) = 0 \quad \text{and} \quad f'(-1) = 0. \] Differentiating \( f(x) \): \[ f'(x) = 5ax^4 + 4bx^3 - 15x^2. \] Substituting \( x = 1 \): \[ 5a + 4b - 15 = 0. \] Substituting \( x = -1 \): \[ 5a - 4b - 15 = 0. \]
Step 3: Determine the values of \( a \) and \( b \).
Adding the two equations: \[ (5a + 4b - 15) + (5a - 4b - 15) = 0 \] \[ 10a - 30 = 0 \] \[ a = 3. \] Substituting \( a = 3 \) into the equation: \[ 5a + 4b - 15 = 0, \] we get \[ 15 + 4b - 15 = 0 \] \[ 4b = 0 \] \[ b = 0. \] Therefore, the polynomial becomes: \[ f(x) = 3x^5 - 5x^3. \]
Step 4: Evaluate \( f(2) - f(-2) \).
First, calculate \( f(2) \): \[ f(2) = 3(2)^5 - 5(2)^3 = 3 \times 32 - 5 \times 8 = 96 - 40 = 56. \] Next, calculate \( f(-2) \): \[ f(-2) = 3(-2)^5 - 5(-2)^3 = 3(-32) - 5(-8) = -96 + 40 = -56. \] Therefore, \[ f(2) - f(-2) = 56 - (-56) = 112. \]
Step 5: Conclusion.
Thus, after determining the polynomial using the limit and extrema conditions, the required value is obtained by direct substitution.
Final Answer: \( 112 \)