A condition that may have no extremum is:

Explanation: Solution 1. Conceptual Background: A function has no extremum (maximum or minimum) if its derivative is always greater than zero ( ) or always less than zero for all . For a quadratic equation to be true for all real , the leading coefficient must be positive ( ) and the discriminant must be negative ( , where ). 2. Finding the Derivative: Let the given function be: We find the first derivative with respect to : 3. Applying the Condition for No Extremum: For the function to have no extremum, we require: This quadratic inequality holds for all real only if its discriminant is less than zero: 4. Final Simplification: Divide the entire inequality by : Correct Option: 3 ( )

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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:

B.
$b^2 < 3b$

C.
$a^2 < 3b$
(Correct Answer)

D.
$b^2 \geq 3b$

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 ( $a^2 < 3b$ )