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Q6(d):
Find the intervals in which the following functions are strictly increasing or decreasing:
(d) $6 – 9x – x^2$
Solution :
Given Function
Let the function be denoted by $f(x)$:
$f(x) = 6 - 9x - x^2$
Step 1: Differentiation of the Function
To determine the intervals of monotonicity, we compute the first derivative with respect to $x$ [Per the Power Rule of differentiation, $\frac{d}{dx}(x^n) = nx^{n-1}$]:
$f'(x) = \frac{d}{dx}(6) - \frac{d}{dx}(9x) - \frac{d}{dx}(x^2)$
$f'(x) = 0 - 9 - 2x$
$f'(x) = -(9 + 2x)$
Step 2: Finding Critical Points
The critical points occur where $f'(x) = 0$ [By the First Derivative Test criteria for critical values]:
$-(9 + 2x) = 0$
$9 + 2x = 0$
$2x = -9$
$x = -4.5$
Step 3: Interval Analysis
The critical point $x = -4.5$ divides the real number line $\mathbb{R}$ into two disjoint intervals: $(-\infty, -4.5)$ and $(-4.5, \infty)$. We analyze the sign of $f'(x) = -(9 + 2x)$ in these intervals:
| Interval | Test Point ($c$) | Value of $f'(c)$ | Sign of $f'(x)$ | Conclusion |
|---|---|---|---|---|
| $(-\infty, -4.5)$ | $-5$ | $-(9 + 2(-5)) = -(9 - 10) = 1$ | $f'(x) > 0$ | Strictly Increasing |
| $(-4.5, \infty)$ | $0$ | $-(9 + 2(0)) = -9$ | $f'(x) < 0$ | Strictly Decreasing |
Geometric Interpretation
Justification
By the Monotonicity Theorem:
- A function is strictly increasing on an interval $I$ if $f'(x) > 0$ for all $x \in I$.
- A function is strictly decreasing on an interval $I$ if $f'(x) < 0$ for all $x \in I$.
Final Solution: The function is strictly increasing on $(-\infty, -4.5)$ and strictly decreasing on $(-4.5, \infty)$.
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(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.
- 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
CBSE Solutions for Class 12 Mathematics Application of Derivatives
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