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Easy

Question

f space left parenthesis x right parenthesis equals t a n to the power of negative 1 end exponent invisible function application x minus x is decreasing in

  1. left parenthesis negative infinity comma infinity right parenthesis    
  2. left parenthesis 0 comma infinity right parenthesis only    
  3. left parenthesis negative 11 right parenthesis only    
  4. left parenthesis negative infinity comma 0 right parenthesis only    

hintHint:

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

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


    The given function is f(x) = tan-1x - x
    A function is said to be decreasing when it’s first order derivative is negative
    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) = tan-1x - x
    f apostrophe left parenthesis x right parenthesis space equals fraction numerator 1 over denominator 1 plus x squared end fraction minus 1

f apostrophe left parenthesis x right parenthesis space less than space 0
rightwards double arrow fraction numerator 1 over denominator 1 space plus space x squared end fraction minus 1 space less than space 0
space space space space space space space space space space fraction numerator 1 space minus space 1 space minus x squared over denominator 1 space plus space x squared end fraction less than 0 space space space space space... left curly bracket space m a k i n g space t h e space d e n o m i n a t o r space e q u a l right curly bracket
space space space space space space space space space space space space space space space space space fraction numerator negative x squared over denominator 1 space plus x squared end fraction less than 0
N o w comma space x squared space i s space t h e space p o s t i v e space q u a n t i t y space f o r space a n y space x element of R. space
I t space m e a n s space f o r space a n y space r e a l space v a l u e space o f space x comma space x to the power of 2 space end exponent i s space a l w a y s space p o s i t i v e
space space space space space space space space space space
S o comma space t h e space t e r m space fraction numerator x squared over denominator 1 space plus space x squared end fraction space w i l l space b e space a l w a y s space p o s i t i v e.
    The negative sign make sures that the function is negative i.e. decreasing.
    x can take any value of real number.
    And the set of real numbers extends from -∞ to ∞.
    So, the interval for which given function is decreasing is (-∞, ∞)

    We should know the formulas of differentiation. And, we should know the condition for increasing and decreasing function.

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