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Q6(e):
Find the intervals in which the following functions are strictly increasing or decreasing:
(e) $(x + 1)^3 (x – 3)^3$
Solution :
Given Function and Initial Setup
The function is given by $f(x) = (x + 1)^3 (x - 3)^3$.
To determine the intervals of monotonicity, we must find the first derivative $f'(x)$ and determine its sign across the domain of the function.
Step 1: Differentiation using the Product Rule
Let $f(x) = u(x)v(x)$, where $u(x) = (x + 1)^3$ and $v(x) = (x - 3)^3$.
[By the Product Rule: $\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$]
$f'(x) = 3(x + 1)^2 (x - 3)^3 + (x + 1)^3 \cdot 3(x - 3)^2$
Factor out the common terms $3(x + 1)^2 (x - 3)^2$:
$f'(x) = 3(x + 1)^2 (x - 3)^2 [(x - 3) + (x + 1)]$
$f'(x) = 3(x + 1)^2 (x - 3)^2 [2x - 2]$
$f'(x) = 6(x + 1)^2 (x - 3)^2 (x - 1)$
Step 2: Determining Critical Points
To find the critical points, set $f'(x) = 0$:
$6(x + 1)^2 (x - 3)^2 (x - 1) = 0$
The critical points are $x = -1$, $x = 1$, and $x = 3$.
Step 3: Interval Analysis
The critical points divide the real number line into four intervals: $(-\infty, -1)$, $(-1, 1)$, $(1, 3)$, and $(3, \infty)$.
| Interval | Test Point ($x$) | Sign of $f'(x) = 6(x + 1)^2 (x - 3)^2 (x - 1)$ | Nature of Function |
|---|---|---|---|
| $(-\infty, -1)$ | $-2$ | $6(+)(+)(-3) < 0$ | Strictly Decreasing |
| $(-1, 1)$ | $0$ | $6(+)(+)(-1) < 0$ | Strictly Decreasing |
| $(1, 3)$ | $2$ | $6(+)(+)(1) > 0$ | Strictly Increasing |
| $(3, \infty)$ | $4$ | $6(+)(+)(3) > 0$ | Strictly Increasing |
Step 4: Synthesis of Intervals
Since $(x+1)^2$ and $(x-3)^2$ are always non-negative for all $x \in \mathbb{R}$, the sign of $f'(x)$ is governed primarily by $(x-1)$.
- $f'(x) < 0$ for $x \in (-\infty, -1) \cup (-1, 1)$. Since the function is continuous at $x = -1$, the interval of decrease is $(-\infty, 1)$.
- $f'(x) > 0$ for $x \in (1, 3) \cup (3, \infty)$. Since the function is continuous at $x = 3$, the interval of increase is $(1, \infty)$.
Visual Representation of $f'(x)$ Sign Change
Final Solution: The function is strictly decreasing on $(-\infty, 1]$ and strictly increasing on $[1, \infty)$.
More Questions from Class 12 Mathematics Application of Derivatives EXERCISE 6.2
- Q1: Show that the function given by $f(x) = 3x + 17$ is increasing on R.
- Q2: Show that the function given by $f(x) = e^{2x}$ is increasing on R.
- Q3(a): Show that the function given by $f(x) = \sin x$ is (a) increasing in $(0, \frac{\pi}{2})$
- Q3(b): Show that the function given by $f(x) = \sin x$ is (b) decreasing in $(\frac{\pi}{2}, \pi)$
- Q3(c): Show that the function given by $f(x) = \sin x$ is (c) neither increasing nor decreasing in $(0, \pi)$
- Q4(a): Find the intervals in which the function $f$ given by $f(x) = 2x^2 – 3x$ is (a) increasing
- Q4(b): Find the intervals in which the function $f$ given by $f(x) = 2x^2 – 3x$ is (b) decreasing
- Q5(a): Find the intervals in which the function $f$ given by $f(x) = 2x^3 – 3x^2 – 36x + 7$ is (a) increasing
- Q5(b): Find the intervals in which the function $f$ given by $f(x) = 2x^3 – 3x^2 – 36x + 7$ is (b) decreasing
- Q6(a): Find the intervals in which the following functions are strictly increasing or decreasing: (a) $x^2 + 2x – 5$
- Q6(b): Find the intervals in which the following functions are strictly increasing or decreasing: (b) $10 – 6x – 2x^2$
- Q6(c): Find the intervals in which the following functions are strictly increasing or decreasing: (c) $–2x^3 – 9x^2 – 12x + 1$
- Q6(d): Find the intervals in which the following functions are strictly increasing or decreasing: (d) $6 – 9x – x^2$
- Q7: Show that $y = \log(1 + x) - \frac{2x}{2+x}$, $x > – 1$, is an increasing function of $x$ throughout its domain.
- Q8: Find the values of $x$ for which $y = [x(x – 2)]^2$ is an increasing function.
- Q9: Prove that $y = \frac{4\sin\theta}{(2 + \cos\theta)} - \theta$ is an increasing function of $\theta$ in $[0, \frac{\pi}{2}]$.
- Q10: Prove that the logarithmic function is increasing on $(0, \infty)$.
- Q11: Prove that the function $f$ given by $f(x) = x^2 – x + 1$ is neither strictly increasing nor decreasing on $(– 1, 1)$.
- Q12: Which of the following functions are decreasing on $(0, \frac{\pi}{2})$ ?
- Q13: On which of the following intervals is the function $f$ given by $f(x) = x^{100} + \sin x –1$ decreasing ?
- Q14: For what values of $a$ the function $f$ given by $f(x) = x^2 + ax + 1$ is increasing on $[1, 2]$?
- Q15: Let $I$ be any interval disjoint from $[–1, 1]$. Prove that the function $f$ given by $f(x) = x + \frac{1}{x}$ is increasing on $I$.
- Q16: Prove that the function $f$ given by $f(x) = \log \sin x$ is increasing on $(0, \frac{\pi}{2})$ and decreasing on $(\frac{\pi}{2}, \pi)$.
- Q17: Prove that the function $f$ given by $f(x) = \log |\cos x|$ is decreasing on $(0, \frac{\pi}{2})$ and increasing on $(\frac{3\pi}{2}, 2\pi)$.
- Q18: Prove that the function given by $f(x) = x^3 – 3x^2 + 3x – 100$ is increasing in R.
- Q19: The interval in which $y = x^2 e^{–x}$ is increasing is
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