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Q3(c):
Show that the function given by $f(x) = \sin x$ is
(c) neither increasing nor decreasing in $(0, \pi)$
Solution :
Step 1: Establishing the Derivative Function
To determine the intervals of monotonicity for the function $f(x) = \sin x$, we must examine the sign of its first derivative, $f'(x)$, with respect to $x$.
Given: $f(x) = \sin x$
Differentiating with respect to $x$:
$f'(x) = \frac{d}{dx}(\sin x) = \cos x$ [By the derivative rule for trigonometric functions]
Step 2: Analyzing the Critical Points in the Interval $(0, \pi)$
A function is strictly increasing if $f'(x) > 0$ and strictly decreasing if $f'(x) < 0$. To analyze the behavior of $f'(x) = \cos x$ in the interval $(0, \pi)$, we must identify the point where the derivative changes sign.
Setting $f'(x) = 0$:
$\cos x = 0$
In the interval $(0, \pi)$, the only solution is $x = \frac{\pi}{2}$.
Step 3: Evaluating Monotonicity via Sub-intervals
We partition the interval $(0, \pi)$ into two sub-intervals based on the critical point $x = \frac{\pi}{2}$:
| Sub-interval | Sign of $f'(x) = \cos x$ | Nature of $f(x)$ |
|---|---|---|
| $(0, \frac{\pi}{2})$ | $\cos x > 0$ (Positive) | Strictly Increasing |
| $(\frac{\pi}{2}, \pi)$ | $\cos x < 0$ (Negative) | Strictly Decreasing |
Step 4: Geometric Visualization
Step 5: Synthesis of Results
Per the definition of monotonic functions [Definition of Monotonicity], a function is called increasing on an interval $I$ if $f'(x) \geq 0$ for all $x \in I$. Conversely, it is decreasing if $f'(x) \leq 0$ for all $x \in I$.
- Since $f'(x) > 0$ on $(0, \frac{\pi}{2})$, the function is strictly increasing on this sub-interval.
- Since $f'(x) < 0$ on $(\frac{\pi}{2}, \pi)$, the function is strictly decreasing on this sub-interval.
Because the derivative $f'(x)$ changes sign from positive to negative at $x = \frac{\pi}{2}$, the function does not maintain a consistent monotonic behavior across the entire interval $(0, \pi)$. Therefore, the function cannot be classified as globally increasing or globally decreasing on the specified domain.
Final Solution: Since the derivative $f'(x) = \cos x$ is positive in $(0, \pi/2)$ and negative in $(\pi/2, \pi)$, the function $f(x) = \sin x$ is neither increasing nor decreasing in $(0, \pi)$.
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 $(\frac{\pi}{2}, \pi)$
- 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.
- 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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