NIMCET 2014 — Mathematics PYQ

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Question

NIMCET 2014 — Mathematics PYQ

NIMCET | Mathematics | 2014

A condition that $x^3 + ax^2 + bx + c$ may have no extremum is:

Choose the correct answer:

Explanation

Solution

1. Conceptual Background:

A function $f(x)$ has no extremum (maximum or minimum) if its derivative $f'(x)$ is always greater than zero ( $f'(x) > 0$ ) or always less than zero for all $x$ .
For a quadratic equation $Ax^2 + Bx + C > 0$ to be true for all real $x$ , the leading coefficient must be positive ( $A > 0$ ) and the discriminant must be negative ( $D < 0$ , where $D = B^2 - 4AC$ ).

2. Finding the Derivative:

Let the given function be:

f(x) = x^3 + ax^2 + bx + c \text{}

We find the first derivative $f'(x)$ with respect to $x$ :

f'(x) = \frac{d}{dx}(x^3 + ax^2 + bx + c) \text{}

f'(x) = 3x^2 + 2ax + b \text{}

3. Applying the Condition for No Extremum:

For the function to have no extremum, we require:

3x^2 + 2ax + b > 0 \text{}

This quadratic inequality holds for all real $x$ only if its discriminant $D$ is less than zero:

D = (2a)^2 - 4 \cdot 3 \cdot b < 0 \text{}

4a^2 - 12b < 0 \text{}

4. Final Simplification:

Divide the entire inequality by $4$ :

a^2 - 3b < 0 \text{}

a^2 < 3b \text{}

Correct Option: 3 ( $a^2 < 3b$ )