Download App

APPLICATION OF DERIVATIVES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAPPLICATION OF DERIVATIVES5 Marks
Question
Find the intervals in which $\text{f(x)} = \sin 3x - \cos 3x, 0 < x < \pi.$ is strictly increasing or strictly decreasing.
✓

Answer

$\text{f(x)} = \sin \text{3x} - \cos \text{3x}, 0 < \text{x} < \pi.$
$\text{f'(x)} = 3\cos3\text{x} + 3\sin\text{3x}$
$\text{f'(x)} = 0 \Rightarrow \tan 3\text{x} = -1$
$\Rightarrow\text{x} = \frac{\text{n}\pi}{3} + \frac{\pi}{4}, \text{n}\in\text{Z}$
$\Rightarrow\text{x} = \frac{\pi}{4}, \frac{7\pi}{12}, \frac{11\pi}{12}$
$\text{Intervals are:} \bigg(0,\frac{\pi}{4}\bigg), \bigg(\frac{\pi}{4},\frac{7\pi}{12}\bigg), \bigg(\frac{7\pi}{12}, \frac{11\pi}{12}\bigg),\bigg(\frac{11\pi}{12}, \pi\bigg)$
$\text{f(x) is strictly increasing in} \bigg(0, \frac{\pi}{4}\bigg)\cup\bigg(\frac{7\pi}{12},\frac{11\pi}{12}\bigg)$
$\text{and strictly decreasing in} \bigg(\frac{\pi}{4},\frac{7\pi}{12}\bigg)\cup\bigg(\frac{11\pi}{12},\pi\bigg)$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from APPLICATION OF DERIVATIVES→APPLICATION OF DERIVATIVES — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Show that the function $f : Q \rightarrow Q$, defined by $f(x) = 3x + 5,$ is invertible. Also, find $f^{-1}.$
→
In a game, a man wins Rs 5 for getting a number greater than 4 and loses Rs 1 otherwise, when a fair die is thrown. The man decided to thrown a die thrice but to quit as and when he gets a number greater then 4. Find the expected value of the amount he wins or loses.
→
Differentiate the following functions with respect to x:
$\text{x}^{(\sin\text{x}-\cos\text{x})}+\frac{\text{x}^2-1}{\text{x}^2+1}$
→
A particle moves along the curve $y = x^2 + 2x.$ At what point(s) on the curve are the $x$ and $y$ coordinates of the particle changing at the same rate?
→
Prove that the curves $xy = 4$ and $x^2 + y^2 = 8$ touch each other.
→
The rate of growth of a population is proportional to the number present. If the population of a city doubled in the past 25 year, and the present population is 100000, when will the city have a population of 500000?
→
If $\text{y}^\text{x}=\text{e}^{\text{x}-\text{e}},$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{(1+\log\text{y})^2}{\log\text{y}}$
→
Show that the points (3, 4), (-5, 16) and (5, 1) are collinear.
→
There are three urns A, B, and C. Urn A contains 4 red balls and 3 black balls. urn B contains 5 red balls and 4 black balls. Urn C contains 4 red and 4 black balls. One ball is drawn from each of these urns. What is the probability that 3 balls drawn consists of 2 red balls and a black ball?
→
If $x^y = e^{x –y},$ show that $\frac{\text{dy}}{\text{dx}}=\frac{\text{log x}}{\left\{\text{log(x e)}\right\}^{2}}.$
→