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Let f(x) be a differentiable function for all x\; \epsilon \; R. The derivative of f(x) is an even function. If the period of f\left ( 2x \right ) is 1, then the value of f\left ( 2 \right )+f\left ( 4 \right )-f\left ( 6 \right )-f\left ( 8 \right ) is

Option: 1

(1)\; 1


Option: 2

(2)\; 2


Option: 3

(3)\; 0

 


Option: 4

(4)\; \text {Cannot be determined}


Answers (1)

best_answer

\\f\left ( 2x \right )=f\left ( 2x+1 \right )\\f\left ( x \right )=f\left ( x+\frac{1}{2} \right )

So period of f\left ( x \right ) is \frac{1}{2}

So f\left ( 4 \right )=f\left ( 2+\frac{1}{2}\times 4 \right )

f\left ( 6 \right )=f\left ( 2+\frac{1}{2}\times 8 \right )

f\left ( 8 \right )=f\left ( 2+\frac{1}{2}\times 12 \right )

So f\left ( 2 \right )=f\left ( 4 \right )=f\left ( 6 \right )=f\left ( 8 \right )

So value of f\left ( 2 \right )+f\left ( 4 \right )-f\left ( 6 \right )=f\left ( 8 \right )=0

Posted by

Riya

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