Question
Question: If we have a trigonometric function as \(f(x)=\sin \left[ {{\pi }^{2}} \right]x+\cos \left[ -{{\pi }...
If we have a trigonometric function as f(x)=sin[π2]x+cos[−π2]x , where [ . ] is greatest integer function, then f′(x) is
( a ) 9sinx + 9cosx
( b ) 9cos9x – 10sin10x
( c ) 0
( d ) -1
Solution
First we will find the values of [π2] and [−π2], then substitute the values of [π2] and [−π2] in function f(x)=sin[π2]x+cos[−π2]x. Then using chain rule of differentiation we will find out the value of f ’ ( x ).
Complete step-by-step solution:
Before we start the question, let us see what the greatest integer function is and how do we find the value of the greatest integer function.
Let us see what is the greatest integer function and what are its properties.
Function y = [ x ] is called greatest integer function which means the greatest integer less than or equals to x. also, if n belongs to set of integer, then y = [ x ] = n if n≤x<n+1 that is x∈[n,n+1) .
For example if we put x = -3.1 in y = [ x ], then y = - 4 and if we put x = 0.2 in y = [ x ], then y = 0.
Now, we know that π2=(3.14)2=9.85 and −π2=−(3.14)2=−9.85.
So, value of [π2]=[9.85]=9 and [−π2]=[−9.85]=−10.
So, we can re – write function f(x)=sin[π2]x+cos[−π2]xas f(x)=sin9x+cos(−10)x
Also, cos ( - x ) =cos x.
So, f(x)=sin9x+cos10x
So, f′(x)=dxd(sin9x+cos10x)
We know that, dxd(A(x)+B(x))=dxdA(x)+dxdB(x)
So, dxd(sin9x+cos10x)=dxd(sin9x)+dxd(cos10x)
We know that dxd(sinax)=a⋅cosax and dxd(cosax)=−asinax.
So, dxd(sin9x)=9⋅cos9x and dxd(cos10x)=−10sin10x.
f′(x)=dxd(sin9x+cos10x)=9cos9x10sin10x
Hence, option ( b ) is true.
Note: One must know the definition of greatest integer function and how to find the value of greatest integer function for any real value of x. Always remember that dxd(sinax)=a⋅cosax and dxd(cosax)=−asinax and dxd(A(x)+B(x))=dxdA(x)+dxdB(x).