Q28

3 marksShort AnswerSection C

(a) Find the intervals in $\mathbb{R}$ for which the function $f(x) = x^4 - 2x^2$ is increasing or decreasing.

OR

(b) Find : $\int \frac{2x + 1}{\sqrt{18 - 4x - x^2}} dx$

Q: How do I decide where $f(x) = x^4 - 2x^2$ is increasing?

Differentiate to get $f'(x) = 4x(x-1)(x+1)$, find the zeros $x = -1, 0, 1$, and test the sign of f' in each interval. Where f' is positive, namely $(-1, 0)$ and $(1, \infty)$, the function is increasing; where f' is negative it is decreasing.

Calculus (Application of Derivatives / Integrals)

Increasing/Decreasing Functions and Integration

Official Answer

This question has two alternatives; either one earns full marks.

Option (a) - Monotonicity of $f(x) = x^4 - 2x^2$ :

$f'(x) = 4x^3 - 4x = 4x(x - 1)(x + 1)$ .
Critical points: $x = -1, 0, 1$ .
Sign of f' on each interval:
$(-\infty, -1)$ : $f' < 0$ , so decreasing.
$(-1, 0)$ : $f' > 0$ , so increasing.
$(0, 1)$ : $f' < 0$ , so decreasing.
$(1, \infty)$ : $f' > 0$ , so increasing.
Increasing on $(-1, 0)$ union $(1, \infty)$ ; decreasing on $(-\infty, -1)$ union $(0, 1)$ .

Option (b) - Integral:

Write $2x + 1 = -1(-4 - 2x) - 3$ , where $-4 - 2x$ is the derivative of $18 - 4x - x^2$ .
Also $18 - 4x - x^2 = 22 - (x + 2)^2$ .
Integral = $-2\sqrt{18 - 4x - x^2}$ - $3\sin^{-1}((x + 2)/\sqrt{22})$ + C.

increasing decreasing functionfirst derivative testsign chartcritical pointsintegration by substitutioncompleting the squareinverse sine integralmonotonicity

Marking Scheme

1Option (a): 1 mark for $f'(x) = 4x^3 - 4x$ factored, 1 mark for critical points and sign chart, 1 mark for correct increasing/decreasing intervals.
2Option (b): 1 mark for splitting $2x+1 = -1(-4-2x) - 3$ , 1 mark for completing the square $22 - (x+2)^2$ and integrating the root part to -2 sqrt(...), 1 mark for the $-3\sin^{-1}((x+2)/\sqrt{22})$ term with +C.
3Accept open or closed interval notation at critical points; deduct for missing +C in (b).

Hint

For (a) factor $f'(x) = 4x(x-1)(x+1)$ and build a sign chart. For (b) split $2x+1$ into a multiple of the derivative of $(18-4x-x^2)$ plus a constant, then complete the square.

Quick Oral Answer

For (a), $f'(x) = 4x(x-1)(x+1)$ is positive on $(-1,0)$ and $(1, \infty)$ so f increases there and decreases elsewhere; for (b), split $2x+1$ by the derivative of the quadratic to get $-2\sqrt{18-4x-x^2}$ - 3 sin inverse of $(x+2)/\sqrt{22}$ plus C.

Analysis & Explanation

Both alternatives sit at the heart of Unit 3 calculus but test opposite operations: differentiation for monotonicity and integration by splitting.

Option (a) - Concept:

A function increases where its first derivative is positive and decreases where it is negative. Factoring $f'(x) = 4x(x-1)(x+1)$ exposes three sign-change points; a sign chart across them gives the four monotonic intervals.

Option (b) - Concept:

When the numerator is linear and the denominator is the square root of a quadratic, split the numerator into (a multiple of the derivative of the quadratic) plus (a constant). The first part integrates to 2 sqrt(quadratic); the constant part, after completing the square to $22 - (x+2)^2$ , integrates to an inverse-sine.

Exam trap:

In (a), students forget x = 0 as a critical point and merge intervals wrongly. In (b), a sign slip while matching $-2A = 2$ (giving $A = -1$ ) flips the whole answer.

Real-world link:

Monotonicity analysis identifies where profit, cost or population is rising or falling; the inverse-sine integral form appears in arc-length and probability (normal-curve related) computations.

Common Mistakes

1In (a), omitting $x = 0$ as a critical point and reporting only two intervals instead of four.
2In (b), a sign error when matching coefficients (A should be -1), which corrupts both terms.
3In (b), forgetting the constant of integration +C or leaving the quadratic uncompleted before the inverse-sine step.

Interesting Facts

The function $x^4 - 2x^2$ has a 'W' shape with two symmetric minima at $x = \pm 1$ and a local maximum at $x = 0$ , a classic double-well curve also seen in physics potential-energy models.

The technique of splitting a linear numerator into 'derivative part + constant part' is one of the most frequently examined integration tricks in CBSE Applied and Core Mathematics.

Spotted a mistake or something unclear?

Tell us — we fix reported answers fast.

Frequently Asked Questions

How do I decide where $f(x) = x^4 - 2x^2$ is increasing?

Differentiate to get $f'(x) = 4x(x-1)(x+1)$ , find the zeros $x = -1, 0, 1$ , and test the sign of f' in each interval. Where f' is positive, namely $(-1, 0)$ and $(1, \infty)$ , the function is increasing; where f' is negative it is decreasing.

Why do we split the numerator $2x+1$ in the integral?

Splitting $2x+1$ into a multiple of the derivative of the denominator's quadratic plus a constant turns the problem into two standard integrals: one of the form derivative/sqrt giving 2 sqrt, and one constant/ $\sqrt{a^2 - u^2}$ giving an inverse sine after completing the square.