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The interval of increase of the function f space left parenthesis x right parenthesis equals x minus e to the power of x plus t a n invisible function application open parentheses fraction numerator 2 pi over denominator 7 end fraction close parentheses is

  1. left parenthesis 0 comma infinity right parenthesis    
  2. left parenthesis negative infinity comma 0 right parenthesis    
  3. left parenthesis 1 comma infinity right parenthesis    
  4. left parenthesis negative 1 , 1 right parenthesis    

Hint:

We are given a function. We have to find the interval for which it is increasing. We will take the first order derivative to find the answer

The correct answer is: left parenthesis negative infinity comma 0 right parenthesis


    The given function is f open parentheses x close parentheses equals x minus e to the power of x plus tan open parentheses fraction numerator 2 pi over denominator 7 end fraction close parentheses
    A function is said to be increasing when it’s first order derivative is positive.
    It means f’(x) > 0
    We will use this condition to find the answer.
    Taking the first order derivative of the given function.
    f(x) = x minus e to the power of x plus tan open parentheses fraction numerator 2 pi over denominator 7 end fraction close parentheses
    f’(x) = 1 – ex + 0

    Now for the function to be increasing,
    f'(x) > 0
    1 - ex > 0
    1 > ex
    e< 1
    Taking loge of both the sides. It means we are taking natural log on both the sides.
    xlogee < loge1
    x < 0           ...{ logee = 1 and loge1 = 0}
    x can have any value less than zero.
    There is no lower limit on x.
    So, the interval will be (-∞,0)

    For such questions, we should know different formulas of differentiation.

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