recategorized by
415 views
0 votes
0 votes

The function $f(x)=\sin x(1+ \cos x)$ which is defined for all real values of $x$

  1. has a maximum at $x= \pi /3$
  2. has a maximum at $x= \pi$
  3. has a minimum at $x= \pi /3$
  4. has neither a maximum nor a minimum at $x=\pi/3$
recategorized by

1 Answer

1 votes
1 votes

$\underline{\textbf{Answer:}\Rightarrow\;\mathbf A}$

$\underline{\textbf{Explanation:}\Rightarrow }$

Given: $f(x) = \sin x(1+\cos x)$

On differentiating the above expression  w.r.t. $\mathrm x$, we get:

$f'(x) = \sin x (1+\cos x)+\sin x(-\sin x) = \cos x + \cos^2 x-\sin ^2x = \cos x + \cos 2x$

Equate the above expression with $0$:

$\implies \cos x = -\cos 2x$

This is equal when $x = \frac{\pi}{3}$

So, this is the critical point.

Now, for knowing whether its maxima or minima, further differentiate $f{''}(x)$, we get:

$f^{''}(x) = -\sin x - 2\sin 2x = -(\sin x + 2\sin 2x)$

$\implies f^{''}(\frac{\pi}{3}) = -(\frac{\sqrt 3}{2} + 2.\frac{\sqrt 3}{2}) \lt 0$

$\therefore$ It's a point of Maxima.

$\therefore \mathbf A$ is the correct option.

edited by

Related questions

3 votes
3 votes
1 answer
1
Arjun asked Sep 23, 2019
497 views
It is given that $e^a+e^b=10$ where $a$ and $b$ are real. Then the maximum value of $(e^a+e^b+e^{a+b}+1)$ is$36$$\infty$$25$$21$
1 votes
1 votes
1 answer
2
Arjun asked Sep 23, 2019
459 views
Suppose that the function $h(x)$ is defined as $h(x)=g(f(x))$ where $g(x)$ is monotone increasing, $f(x)$ is concave, and $g’’(x)$ and $f’’(x)$ exist for all $x$....
1 votes
1 votes
1 answer
3
Arjun asked Sep 23, 2019
548 views
The function $f(x) = x^{1/x}, \: x \neq 0$ hasa minimum at $x=e$;a maximum at $x=e$;neither a maximum nor a minimum at $x=e$;None of the above
0 votes
0 votes
0 answers
4
Arjun asked Sep 23, 2019
388 views
Let $f(x)=\sin x^2, \: x \in \mathbb{R}$. Then$f$ has no local minima$f$ has no local maxima$f$ has local minima at $x=0$ and $x=\pm\sqrt{(k+\frac{1}{2} ) \pi}$ for odd i...