Question

If $f(x) = \sin \left[ {{\pi ^2}} \right]x + \cos \left[ { - {\pi ^2}} \right]x$ then $f'(x)$ is , here $\left[ {{\pi ^2}} \right]$ and $\left[ { - {\pi ^2}} \right]$ greatest integer function not greater than its value

Choose the correct option.

A. $\sin 9x + \cos 9x$

B. $9\cos 9x - 10\sin 10x$

C. $0$

D. $ - 1$

Answer 1

Hint: For solving this particular question which involves greatest integer function. We must know that for $\left[ x \right]$, the value of the greatest integer function is given by the larger value of the integer whose value is less than or equal to $x$.

Complete step by step solution:

It is given that $f(x) = \sin \left[ {{\pi ^2}} \right]x + \cos \left[ { - {\pi ^2}} \right]x$ , where $\left[ {{\pi ^2}} \right]$ and $\left[ { - {\pi ^2}} \right]$ greatest integer function not greater than its value.

Let us take given equation,

\[f(x) = sin[{\pi ^2}]x + cos[ - {\pi ^2}]x\]

As we know, the value of $\pi $ is $3.14$ .

Therefore , the value of ${\pi ^2}$ is $9.86$ .

Therefore, we get ,

\[ \Rightarrow [{\pi ^2}] = 9 \text{ and }[ - {\pi ^2}] = - 10\]

As we know [ ] denotes the greatest integer function not greater than its value.

Now, put this result in the given equation,

\[ \Rightarrow f(x) = sin9x + cos( - 10)x\]

We know that $\cos ( - x) = \cos x$ . therefore,

$ = sin9x + cos10x$

Now differentiate the above equation with respect to $x$.

\[ \Rightarrow f^\prime (x) = 9cos9x - 10sin10x\]

Hence , we get the required result .

Therefore, we can say that option (B) is correct .

Note:
The other name for greatest integer function is floor function , the name floor is given because the graph we obtain from the greatest integer function looks like a step.

Greatest integer function is represented by using a square bracket that is [ ] .

For $\left[ x \right]$ , the value of the greatest integer function is given by the larger value of the integer whose value is less than or equal to $x$ .

Differentiation of $\sin x$ is $\cos x$ and $\cos x$ is minus $\sin x$.

Differentiation of $\sin ax$ is $a\cos x$ and $\cos ax$ is minus $a\sin x$.

Greatest integer function will always give an integer value.