Show that f(x)= is increasing on (0, 2 ) and decreasing on ( 2 , … - Vidyadip

Question

Show that $\text{f}(\text{x})=\log\sin\text{x}$ is increasing on $\Big(0,\frac{\pi}{2}\Big)$ and decreasing on $\Big(\frac{\pi}{2},\pi\Big).$

✓

Answer

$\text{f}(\text{x})=\log\sin\text{x}$
$\text{f}'(\text{x})=\frac{1}{\sin\text{x}}\cos\text{x}=\cot\text{x}$
In interval $\Big(0,\frac{\pi}{2}\Big),\text{f}'(\text{x})=\cot\text{x}>0.$
$\therefore$ f is strictly increasing in $\Big(0,\frac{\pi}{2}\Big).$
In interval $\Big(\frac{\pi}{2},\pi\Big),\text{f}'(\text{x})=\cot\text{x}<0.$
$\therefore$ f is strictly decreasing in $\Big(\frac{\pi}{2},\pi\Big).$

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 "3 Marks" 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

If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability $\frac{1}{2}$ ).

Find the matrix A such that
$ \begin{bmatrix}4\\1\\3\end{bmatrix}\text{A}=\begin{bmatrix}-4&8&4\\-1&2&1\\-3&6&3\end{bmatrix}$

Evalute the following integrals:
$\int\frac{1}{\text{x}(3+\log\text{x})}\text{dx}$

If $\text{A}=\begin{bmatrix}0&-1&2\\4&3&-4\end{bmatrix}$ and $\text{B}=\begin{bmatrix}4&0\\1&3\\2&6\end{bmatrix},$ then verify that $(\text{AB})'=\text{B}'\text{A}'.$

Evaluate the following integrals:
$\int\text{e}^{\text{x}}\Big(\frac{1}{\text{x}^2}-\frac{2}{\text{x}^3}\Big)\text{dx}$

In each of the form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y² = a (b² – x²)

Prove that the function f : N → N, defined by f(x) = x² + x + 1, is one-one but not onto.

Find the angle between two vectors $\hat i - 2\hat j + 3\hat k$ and $3\hat i - 2\hat j + \hat k\;$

If $\text{A}=\begin{bmatrix}4&-1&-4\\3&0&-4\\3&-1&-3\end{bmatrix},$ show that A² = I₃.

Evaluate the definite integral in Exercise:
$\int_{0}^{2}\frac{6\text{x}+3}{\text{x}^{2}+4}\text{dx}$