Question

For a real number 'a', if a real valued function $f(x) = 4x^3 + ax^2 + 3x - 2$ is monotonic in its domain, then the range of 'a' is

• A. (-6,6)
• B. Empty set
• C. (-2,2)
• D. (2,4)

Hint:

A cubic function is monotonic over its entire domain if its derivative (a quadratic function) has a discriminant $\Delta \leq 0$. This ensures the derivative never crosses the x-axis, so it doesn't change sign.

Answer 1

The Correct Option is A

A function is monotonic if its derivative, $f'(x)$, does not change sign. That is, either $f'(x) \geq 0$ for all $x$, or $f'(x) \leq 0$ for all $x$.
First, find the derivative of the function $f(x) = 4x^3 + ax^2 + 3x - 2$.
$f'(x) = 12x^2 + 2ax + 3$.
This derivative is a quadratic function of $x$. The graph of $y=f'(x)$ is a parabola.
Since the coefficient of the $x^2$ term (which is 12) is positive, the parabola opens upwards.
For such a parabola to be always non-negative ($f'(x) \geq 0$), it must either touch the x-axis at exactly one point (one real root) or stay entirely above the x-axis (no real roots).
This condition means that the discriminant ($\Delta$) of the quadratic equation $12x^2 + 2ax + 3 = 0$ must be less than or equal to zero.
$\Delta = B^2 - 4AC \leq 0$.
Here, $A=12$, $B=2a$, and $C=3$.
$(2a)^2 - 4(12)(3) \leq 0$.
$4a^2 - 144 \leq 0$.
$4a^2 \leq 144$.
$a^2 \leq 36$.
This inequality is satisfied when $-6 \leq a \leq 6$.
The range of values for 'a' is the closed interval $[-6, 6]$.
The given options are open intervals. The option that most closely represents this range is $(-6, 6)$. In multiple-choice tests, it's common for an open interval to be provided when the correct answer is a closed interval.