Question

Given that \[|\sin 4x + 3|\]. Then what will the maximum and minimum values of the given function respectively?
(a) $4,2$
(b) $1,4$
(c) $2,4$
(d) $3,6$

Answer 1

Hint: The given problem revolves around the concepts of trigonometry. So, we will use the condition of existence for the given ‘sine’ term i.e. $ - 1 \leqslant \sin z \leqslant + 1$. Where the ‘sine’ function exists between these values and then adding the $3$ in this entire function the desired solution is obtained.

Complete step-by-step solution:
Since, we have given the function ,
$|\sin x + 3|$, where mode ‘| |’ sign represents the positive solution of the function because any trigonometric term or any angle cannot be negative, ........................(i)
Now, let us assume that $f(x)$ is functionally active for the given data or say, problem
$ \Rightarrow f(x) = |\sin 4x + 3|$
As a result, of the function ‘$\sin z$’ we know that the values exists between $ - 1$ and $ + 1$ respectively
(Where, ‘$z$’ is noted as $4x$ implies with the given problem)
Hence, $ - 1$ represents ‘minima’ or ‘minimum’ value and $ + 1$ holds the ‘maxima’ or ‘maximum’ value of the function $\sin z$.......................................… (ii)
Therefore, we can write the above function '$f(x)$’ in the form of
$ \Rightarrow - 1 \leqslant \sin z \leqslant + 1$ Or, $ = - 1 \leqslant \sin 4x \leqslant + 1$
Or,
$ \Rightarrow f(x) = - 1 \leqslant \sin z \leqslant + 1$ Or, $f(x) = - 1 \leqslant \sin 4x \leqslant + 1$
Now, adding $3$ in above function, we get
$ \Rightarrow f(x) = ( - 1 + 3) \leqslant |\sin 4x + 3| \leqslant ( + 1 + 3)$ … [From (i)]
Solving the equation systematically, we get
$ \Rightarrow f(x) = 2 \leqslant |\sin 4x + 3| \leqslant 4$
From (ii),
$ \Rightarrow $Minima$ = 2$ and,
Maxima$ = 4$ respectively
$\therefore $Hence, the option (a) is correct.

Note: One should know the condition of trigonometric terms such as ‘sine’, ‘cosine’, etc. while solving a question (here, we used the condition of ‘sine’ term). We should also know the basic concepts of simplification of the problem studied in earlier classes. As a result, to get the accurate answer we must take care of the calculations so as to be sure of our final answer.