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Q2:
Show that the function given by $f(x) = e^{2x}$ is increasing on R.
Solution :
Step 1: Establishing the Function and Derivative
To determine the monotonicity of the function $f(x) = e^{2x}$ on the set of real numbers $\mathbb{R}$, we must examine the sign of its first derivative $f'(x)$.
The given function is: $$f(x) = e^{2x}$$ [By the Chain Rule of differentiation, where $\frac{d}{dx}[e^{u(x)}] = e^{u(x)} \cdot u'(x)$]: $$f'(x) = \frac{d}{dx}(e^{2x}) = e^{2x} \cdot \frac{d}{dx}(2x)$$ $$f'(x) = e^{2x} \cdot 2 = 2e^{2x}$$
Step 2: Analysis of the Derivative
We analyze the behavior of $f'(x) = 2e^{2x}$ for all $x \in \mathbb{R}$.
- The exponential function $e^u$ is strictly positive for all real values of $u$ ($e^u > 0, \forall u \in \mathbb{R}$).
- Since $x \in \mathbb{R}$, it follows that $2x \in \mathbb{R}$.
- Therefore, $e^{2x} > 0$ for all $x \in \mathbb{R}$.
- Multiplying by the constant $2$ (which is positive) preserves the inequality: $2e^{2x} > 0$.
Thus, $f'(x) > 0$ for all $x \in \mathbb{R}$.
Step 3: Visual Representation of $f(x) = e^{2x}$
Step 4: Theoretical Conclusion
According to the First Derivative Test for Monotonicity:
- If $f'(x) > 0$ for all $x$ in an interval $I$, then the function $f$ is strictly increasing on that interval. [Definition of Strictly Increasing Functions].
Final Solution: Since $f'(x) = 2e^{2x} > 0$ for all $x \in \mathbb{R}$, the function $f(x) = e^{2x}$ is strictly increasing on $\mathbb{R}$.
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.
- 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.
- 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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