Prove that the function given by $f(x) = x^3 - 3x^2 + 3x - 100$ is increasing in $\mathbf{R}$.

Prove that the function given by f(x) = x³ - 3x² + – Class 12 Mathematics

Exercise 6.2 · Q18

f(x) = x^3 - 3x^2 + 3x - 100

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Concept understanding — Monotonic Function Analysis

Monotonic Function Analysis

A function is monotonic on an interval when it moves in a single direction across that interval — either always rising or always falling, never doubling back. Derivatives give us a clean, mechanical way to detect this, which is why monotonicity is one of the first applications of the derivative.

Increasing, Decreasing, Monotonic

On an interval $I$ , a function $f$ is:

increasing if $x_1 < x_2 \Rightarrow f(x_1) \le f(x_2)$ ,
strictly increasing if $x_1 < x_2 \Rightarrow f(x_1) < f(x_2)$ ,
decreasing if $x_1 < x_2 \Rightarrow f(x_1) \ge f(x_2)$ ,
strictly decreasing if $x_1 < x_2 \Rightarrow f(x_1) > f(x_2)$ .

A function that is either increasing throughout $I$ or decreasing throughout $I$ is called monotonic on $I$ .

The Derivative Test

The slope of the tangent tells you the direction of travel. If $f$ is differentiable on an open interval $I$ :

f'(x) > 0 \text{ on } I \implies f \text{ is strictly increasing on } I

The idea is intuitive: a positive slope means the graph climbs as you move right, a negative slope means it falls.

How to Analyse Monotonicity

Compute $f'(x)$ .
Solve $f'(x) = 0$ (and note where $f'$ is undefined). These critical points split the domain into intervals.
Test the sign of $f'$ in each interval.
Read off where $f$ increases ( $f' > 0$ ) and decreases ( $f' < 0$ ).

Example. For $f(x) = x^2 - 4x + 1$ , $f'(x) = 2x - 4$ . So $f'(x) < 0$ for $x < 2$ and $f'(x) > 0$ for $x > 2$ : the function decreases on $(-\infty, 2)$ and increases on $(2, \infty)$ . …

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$f'(x) = 3(x-1)^2 \ge 0$ for all real $x$ , so $f$ is increasing on $\mathbf{R}$ .

To test monotonicity we examine the sign of the derivative: if $f'(x) \ge 0$ throughout an interval (with equality only at isolated points), then $f$ is increasing there.

1. Differentiate.

f'(x) = 3x^2 - 6x + 3 = 3(x^2 - 2x + 1) = 3(x-1)^2.

2. Determine the sign.

Since $(x-1)^2 \ge 0$ for every real $x$ ,

f'(x) = 3(x-1)^2 \ge 0 \quad \text{for all } x \in \mathbf{R},

with equality only at the single point $x = 1$ .

3. Conclude. …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

AHSEC Higher Secondary (HS) Final Examination 2025Set ANNUAL4 marks
$f$
(a) increasing
(b) decreasing. OR A balloon, which always remains spherical, has a variable diameter $(2x+1)$ . Find the rate of change of its volume with respect to $x$ .
›Reveal solutionSolution
Main: sign of $f'(x)=12(x-3)(x+2)$ splits the real line at $x=-2,3$ . OR: express volume in terms of $x$ and differentiate.

Main: $f(x)=4x^3-6x^2-72x+30$ .

$f'(x)=12x^2-12x-72=12(x^2-x-6)=12(x-3)(x+2)$

Set $f'(x)=0$ : $x=-2,\ x=3$ . These split the real line into three intervals:

$x<-2$ : both $(x-3)$ and $(x+2)$ negative $\implies$ product positive $\implies f'(x)>0$ (increasing)

$-2<x<3$ : $(x-3)$ negative, $(x+2)$ positive $\implies$ product negative $\implies f'(x)<0$ (decreasing)

$x>3$ : both positive $\implies f'(x)>0$ (increasing)

So $f$ is increasing on $(-\infty,-2)\cup(3,\infty)$ and decreasing on $(-2,3)$ .

OR: A spherical balloon has diameter $2x+1$ , so radius $r=\dfrac{2x+1}{2}$ .

…
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AHSEC Higher Secondary (HS) Final Examination 2020Set ANNUAL4 marks
$f(x) = x^3 - 6x^2 - 36x + 7$
(i) strictly increasing
(ii) strictly decreasing. OR Find the area of the largest rectangle that can be formed having a perimeter of 40 meters.
›Reveal solutionSolution
Find $f'(x)$ , factor it, and study its sign; for the OR, maximize area subject to the fixed-perimeter constraint using calculus.

$f(x)=x^3-6x^2-36x+7$

$f'(x)=3x^2-12x-36=3(x^2-4x-12)=3(x-6)(x+2)$

Critical points: $x=6,-2$ . Sign of $f'(x)=3(x-6)(x+2)$ on each interval:

$x<-2$ : both factors negative $\Rightarrow$ product positive $\Rightarrow f'>0$ (increasing)

$-2<x<6$ : $(x+2)>0,(x-6)<0\Rightarrow f'<0$ (decreasing)

$x>6$ : both positive $\Rightarrow f'>0$ (increasing)

So $f$ is strictly increasing on $(-\infty,-2)\cup(6,\infty)$ and strictly decreasing on $(-2,6)$ .

OR: largest rectangle of perimeter 40 m

Let sides be $l,b$ . Perimeter: $2(l+b)=40\Rightarrow b=20-l$ .

Area: $A(l)=l(20-l)=20l-l^2$

…
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Q.Prove that the function given by $f(x) = x^3 - 3x^2 + 3x - 100$ is increasing in $\mathbf{R}$ .

Concept understanding — Monotonic Function Analysis

Monotonic Function Analysis

Increasing, Decreasing, Monotonic

The Derivative Test

How to Analyse Monotonicity

Method: Proving a Function Is Increasing Everywhere Using a Perfect-Square Derivative

Steps

Common Mistakes

Q.Prove that the function given by f(x)=x3−3x2+3x−100f(x) = x^3 - 3x^2 + 3x - 100f(x)=x3−3x2+3x−100 is increasing in R\mathbf{R}R.

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Q.Prove that the function given by $f(x) = x^3 - 3x^2 + 3x - 100$ is increasing in $\mathbf{R}$ .

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