Q.The interval on which the function $f(x) = 2x^3 + 9x^2 + 12x - 1$ is decreasing is:
(A) $[-1, \infty)$
(B) $[-2, -1]$
(C) $(-\infty, -2]$
(D) $[-1, 1]$

Concept: Derivative Sign Analysis — a function is decreasing where its derivative is negative.

Step 1: Differentiate.

$f'(x) = 6x^2 + 18x + 12 = 6(x^2 + 3x + 2) = 6(x+1)(x+2)$.

Step 2: Find critical points: $x = -2$ and $x = -1$.

Step 3: Test sign of $f'(x)$ on the intervals:

• For $x < -2$, say $x = -3$: $( - )( - ) = +$, so $f'(x) > 0$ (increasing).
• For $-2 < x < -1$, say $x = -1.5$: $( - )( + ) = -$, so $f'(x) < 0$ (decreasing). …

The key idea is simple: a function is decreasing on intervals where its slope — the derivative — is negative. So the entire problem reduces to finding where $f'(x) < 0$.

1. Find the derivative

$f(x) = 2x^3 + 9x^2 + 12x - 1$

Differentiate term by term:

This is a quadratic. To analyse its sign, we first factor it.

2. Factor the derivative

Take out the common factor 6:

Now factor the quadratic inside:

So:

3. Find the critical points

Set $f'(x) = 0$:

These are the points where the slope is zero — possible locations of local maxima, minima, or plateaus. They split the real line into three intervals:

4. Test the sign of $f'(x)$ on each interval

Since $6 > 0$, the sign of $f'(x)$ is the same as the sign of $(x+1)(x+2)$.

Pick a test point in each interval:

• Interval $(-\infty, -2)$: take $x = -3$

  $(x+1)(x+2) = (-2)(-1) = 2 > 0$ → $f'(x) > 0$ → function is increasing.

• Interval $(-2, -1)$: take $x = -1.5$

  $(x+1)(x+2) = (-0.5)(0.5) = -0.25 < 0$ → $f'(x) < 0$ → function is decreasing.

• Interval $(-1, \infty)$: take $x = 0$

  $(x+1)(x+2) = (1)(2) = 2 > 0$ → $f'(x) > 0$ → function is increasing. …

Method: Finding the Interval Where a Function is Decreasing

Applies to any question that gives a polynomial (or other differentiable) function and asks for the interval(s) on which it is decreasing (or increasing).

Steps

Step 1: Differentiate the function

Find $f'(x)$.

Step 2: Factor the derivative completely

Pull out any common numerical factor, then factor the remaining polynomial (quadratic, etc.) into linear factors — this makes the sign of $f'(x)$ easy to read off.

Step 3: Mark the roots on a number line to split the domain into intervals

The roots of $f'(x)=0$ are the only places the sign of $f'$ can change.

Step 4: Test the sign of $f'(x)$ in each interval …

Common Mistakes

Mistake 1: Making a sign error while testing a factored expression

Why it's wrong: with a product like $(x+1)(x+2)$, miscounting the number of negative factors at a test point (e.g. treating $(-1.5+1)$ as positive instead of negative) flips the conclusion about whether the function is increasing or decreasing on that interval. Correct approach: evaluate each factor's sign separately at the test point before multiplying them together.

Mistake 2: Forgetting to factor out the leading numerical coefficient before analysing sign …