Q.A function 𝑓(𝑥) = 10 − 𝑥 − 2𝑥2 is increasing on the interval
(A) (−∞, − 1/4]
(B) (−∞, 1/4)
(C) [− 1/4, ∞)
(D) [− 1/4, 1/4]

The key idea is the Increasing Function Test: a differentiable function is increasing where its derivative is non-negative ($f'(x) \geq 0$).

Step 1: Differentiate $f(x) = 10 - x - 2x^2$.

Step 2: Set $f'(x) \geq 0$ for increasing behaviour.

$-1 - 4x \geq 0 \quad \Rightarrow \quad -4x \geq 1 \quad \Rightarrow \quad x \leq -\frac{1}{4}$ …

The key idea is the Increasing Function Test: a differentiable function $f$ is increasing on an interval if its derivative $f'(x) \ge 0$ for all $x$ in that interval. This is not a trick — it’s the direct definition of what “increasing” means in calculus: the slope of the tangent must be non-negative.

For a quadratic like this, the derivative is linear, so the inequality is simple to solve. Let’s work through it.

1. Find the derivative.

   $f(x) = 10 - x - 2x^2$

   Differentiate term by term:

   $f'(x) = 0 - 1 - 4x = -1 - 4x$

2. Set up the increasing condition.

   We need $f'(x) \ge 0$:

   $-1 - 4x \ge 0$

3. Solve the inequality.

   Add $1$ to both sides: $-4x \ge 1$

   Divide by $-4$ (remember: dividing by a negative flips the inequality sign):

   $x \le -\frac{1}{4}$

   So $f$ is increasing for all $x$ less than or equal to $-\frac14$.

4. Interpret the result. …

Method: Determining Where a Function Is Increasing (or Decreasing)

This is the standard approach for any question that asks you to find the interval(s) on which a function is increasing or decreasing, or to identify which of several given intervals is correct.

Steps

Step 1: Differentiate the function

Find $f'(x)$ using the standard differentiation rules (power rule, etc.). This derivative tells you the slope of the tangent at every point.

Step 2: Set up the correct inequality

• For increasing (non-decreasing): solve $f'(x) \geq 0$.
• For strictly increasing: solve $f'(x) > 0$.
• For decreasing: solve $f'(x) \leq 0$.

This follows directly from the Increasing/Decreasing Function Test: the sign of the derivative tells you the direction the function is moving.

Step 3: Solve the inequality for $x$

Since $f'(x)$ is usually linear or quadratic here, solving the inequality is routine algebra. Remember: multiplying or dividing an inequality by a negative number flips its direction.

$f'(x) \gtrless 0 \quad\Longrightarrow\quad \text{solve for the interval of } x$ …

Common Mistakes

Mistake 1: Forgetting to flip the inequality sign when dividing by a negative number

Solving $-1-4x\ge 0$ gives $-4x\ge 1$; dividing both sides by $-4$ must flip the inequality to $x\le -\frac14$. A student who forgets this rule gets $x\ge -\frac14$, which is exactly the decreasing interval, not the increasing one — and would wrongly match a different option.

Mistake 2: Excluding the point where the derivative is zero …