Show that the function f(x)= (2x+ 4 ) is decreasing on (3 8 ,5 8 … - Vidyadip

Question

Show that the function $\text{f}(\text{x})=\sin\Big(2\text{x}+\frac{\pi}{4}\Big)$ is decreasing on $\Big(\frac{3\pi}{8},\frac{5\pi}{8}\Big).$

✓

Answer

$\text{f}(\text{x})=\sin\Big(2\text{x}+\frac{\pi}{4}\Big)$
$\text{f}'(\text{x})=2\cos\Big(2\text{x}+\frac{\pi}{4}\Big)$
Here,
$\frac{3\pi}{8}<\text{x}<\frac{5\pi}{8}$
$\Rightarrow\frac{3\pi}{4}<2\text{x}<\frac{5\pi}{4}$
$\Rightarrow\pi<2\text{x}+\frac{\pi}{4}<\frac{3\pi}{2}$
$\Rightarrow\cos\Big(2\text{x}+\frac{\pi}{4}\Big)<0$ $[\because$ Cos function is negative in third quadrent$]$
$\Rightarrow2\cos\Big(2\text{x}+\frac{\pi}{4}\Big)<0$
$\Rightarrow\text{f}'(\text{x})<0,\ \forall\ \text{x}\in\Big(\frac{3\pi}{8},\frac{5\pi}{8}\Big)$
So, f(x) is decreasing on $\Big(\frac{3\pi}{8},\frac{5\pi}{8}\Big).$

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 Increasing and Decreasing Functions→Increasing and Decreasing Functions — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Verify Rolle's theorem for the following function on the indicated intervals
$f(x) = x^2 -4x + 3$ on $[1, 3]$

If $\vec{\text{a}}\text{ and }\vec{\text{b}}$ are two non-collinear vectors having the same initial point. What are the vectors represented by $\vec{\text{a}}+\vec{\text{b}}\text{ and }\vec{\text{a}}-\vec{\text{b}}$.

By computing the shortest distance determine whether the following pairs of lines intersect or not:
$\vec{\text{r}}=\big(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}\big)+\lambda\big(3\hat{\text{i}}-\hat{\text{j}}\big)$ and $\vec{\text{r}}=\big(4\hat{\text{i}}-\hat{\text{k}}\big)+\mu\big(2\hat{\text{i}}+3\hat{\text{k}}\big)$

Evaluate the following integrals:
$\int\limits^\pi_0\sin^{100}\text{x}\cos^{101}\text{x dx}$

Evaluate: $\int \frac{\sin x}{(1 - \cos x) ( 2 - cos x)} \text{dx}$

Solve $2(\text{y}+3)-\text{xy}\frac{\text{dy}}{\text{dx}}=0,$ given that y(1) = -2.

Find the shortest distance between the following pairs of lines whose vector equation are:
$\vec{\text{r}}=\big(2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}\big)+\lambda\big(2\hat{\text{i}}-5\hat{\text{j}}+2\hat{\text{k}}\big)$ and, $\vec{\text{r}}=\big(\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}\big)+\mu\big(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\big)$

Show that the following systems of linear equations has infinite number of solutions and solve:

2x + y - z = 2,

-x - 2y + 2z = 1

If $\vec{\text{a}} \times \vec{\text{b}} = \vec{\text{c}} \times \vec{\text{d}}$ and $\vec{\text{a}} \times \vec{\text{c}} = \vec{\text{b}} \times \vec{\text{d}},$ show that $\vec{\text{a}} - \vec{\text{d}}$ is parallel to $\vec{\text{b}} - \vec{\text{c}},$ where $\vec{\text{a}} \neq \vec{\text{d}}$ and $\vec{\text{b}} \neq \vec{\text{c}}.$

Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius $5\sqrt{3}\text{cm}$ is $500\pi\text{cm}^{3}$ .