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Q8:
Find the values of $x$ for which $y = [x(x – 2)]^2$ is an increasing function.
Solution :
Step 1: Defining the Function and its Derivative
Given the function: $f(x) = [x(x - 2)]^2$
Expanding the interior term first simplifies the differentiation process: $f(x) = (x^2 - 2x)^2$ [Algebraic Expansion]
Now, calculate the first derivative $f'(x)$ using the Chain Rule: $\frac{d}{dx}[u^n] = n \cdot u^{n-1} \cdot \frac{du}{dx}$ [Chain Rule of Differentiation]
$f'(x) = 2(x^2 - 2x) \cdot \frac{d}{dx}(x^2 - 2x)$
$f'(x) = 2(x^2 - 2x)(2x - 2)$
$f'(x) = 4(x)(x - 2)(x - 1)$
Step 2: Finding Critical Points
To determine the intervals of increase, we set $f'(x) = 0$ to find the critical points:
$4x(x - 1)(x - 2) = 0$
The critical points are $x = 0$, $x = 1$, and $x = 2$.
Step 3: Interval Analysis
The critical points divide the number line into four distinct intervals. We test the sign of $f'(x)$ in each interval:
| Interval | Test Point ($k$) | Sign of $f'(k) = 4k(k-1)(k-2)$ | Behavior |
|---|---|---|---|
| $(-\infty, 0)$ | $-1$ | $4(-1)(-2)(-3) = -24$ | Decreasing |
| $(0, 1)$ | $0.5$ | $4(0.5)(-0.5)(-1.5) = +1.5$ | Increasing |
| $(1, 2)$ | $1.5$ | $4(1.5)(0.5)(-0.5) = -1.5$ | Decreasing |
| $(2, \infty)$ | $3$ | $4(3)(2)(1) = +24$ | Increasing |
Step 4: Visualizing the Function Behavior
Step 5: Conclusion
A function is strictly increasing where $f'(x) > 0$ and increasing where $f'(x) \geq 0$. Based on the interval analysis, $f'(x) > 0$ for $x \in (0, 1) \cup (2, \infty)$.
Final Solution: The function $y = [x(x - 2)]^2$ is an increasing function for $x \in [0, 1] \cup [2, \infty)$.
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) $6 – 9x – x^2$
- Q6(e): Find the intervals in which the following functions are strictly increasing or decreasing: (e) $(x + 1)^3 (x – 3)^3$
- Q7: Show that $y = \log(1 + x) - \frac{2x}{2+x}$, $x > – 1$, is an increasing function of $x$ throughout its domain.
- 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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