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Applications of Derivatives Test - 1

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    Function f(x) = x100 + sin x – 1 is increasing in the interval -

    Solution

    f(x) = x100 + sin x –1 
    Differentiating w.r.t. x, we get 
    f ′(x) = 100 x99 + cos x – 0 
    f ′(x) = 100 x99 + cos x > 0 
    We have to check for every option 
    For A : x > 0 ⇒ φ(x) > φ(0) 
    Let φ(x) = 100x99 + cos x 
    φ(0) = 0 + 1 = 1 
    φ(x) > 0 ⇒ φ(x) > φ(0) 
    ⇒ 100 x99 + cos x > 1 
    ⇒ 100 x99 + cos x > 0

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