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Question: The value of x for which f(x) = \(\left( \sin\frac{\{ x\}}{\lbrack x\rbrack} + \cos\frac{\{ x\}}{\lb...

The value of x for which f(x) = (sin⁡{x}[x]+cos⁡{x}[x])\left( \sin\frac{\{ x\}}{\lbrack x\rbrack} + \cos\frac{\{ x\}}{\lbrack x\rbrack} \right) is maximum ({x} and [x] denotes fractional part and greatest integer part of x respectively)

A

1+π41 + \frac{\pi}{4}

B

2+π42 + \frac{\pi}{4}

C

1–π41–\frac{\pi}{4}

D

None

Answer

1+π41 + \frac{\pi}{4}

Explanation

Solution

Let {x}[x]\frac{\{ x\}}{\lbrack x\rbrack} = a

Ž f (x) = sina + cos a

= 2(sin⁡(π4+α))\sqrt{2}\left( \sin\left( \frac{\pi}{4} + \alpha \right) \right)

f (x) is maximum at a = π4\frac{\pi}{4}

\ {x}[x]\frac{\{ x\}}{\lbrack x\rbrack} = π4\frac{\pi}{4}

Ž {x} = π4\frac{\pi}{4} [x]

It is true at [x] = 1

\ {x} = π4\frac{\pi}{4}

So x = [x] + {x} = 1 + π4\frac{\pi}{4}