Let f be a differentiable function on R satisfying f ′ ( t ) = e t ( cos 2 ⁡ t − s i n 2 t ) and f(0)=1, then which of the following is/are correct ?A) f is bounded in t ∈ ( − ∞ , 0 )B) number of solutions satisfying the equation f ( t ) = e t in [0,2π] is 3C) lim t → 0 f ( t ) ) 1 / t = 1D) f is an even function

Question

Let f be a differentiable function on R satisfying   f      ′    (  t  )  =      e    t    (      cos    2    ⁡  t  −  s  i  n  2  t  ) and f(0)=1, then which of the following is/are correct ?A) f is bounded in   t  ∈  (  −  ∞  ,  0  )B) number of solutions satisfying the equation   f  (  t  )  =      e    t   in [0,2π] is 3C)       lim          t      →      0        f  (  t  )      )          1              /            t        =  1D) f is an even function

Mikayla Cox · Accepted Answer

The correct answers areA f is bounded in   t  ∈  (  −  ∞  ,  0  )B number of solutions satisfying the equation   f  (  t  )  =      e    t   in [0,2π] is 3  f      ′    (  t  )  =      e    t    (      cos    2    ⁡  t  −  sin  ⁡  2  t  )Integrate both the sides with respect to t, we get  f  (  t  )  =      e    t        cos    2    ⁡  t  +  Cf(0)=1⇒C=0  ⇒  f  (  t  )  =      e    t        cos    2    ⁡  t  ⇒  f  (  −  t  )  =      e          −      t            cos    2    ⁡  tClearly, f is neither odd nor even.As   −  ∞  &amp;lt;  t  &amp;lt;  O  ⇒  O  &amp;lt;      e    t    &amp;lt;  1  a  n  d     O  ≤      cos    2    ⁡  t  ≤  1  ⇒  O  &amp;lt;      e    t        cos    2    ⁡  t  &amp;lt;  1Thus, f is bounded in t∈(−∞,0).Then,   f  (  t  )  =      e    t    ⇒      cos    2    ⁡  t  =  1⇒t=0,π,2π∈[0,2π]Therefore,   f  (  t  )  =      e    t   has 3 solutions in [0,2π].Let       lim          t      →      0        (  f  (  t  )      )          1              /            t        =      e    L    (      1          ∞       form)  L  =      lim          t      →      0        (                    e        t                    cos        2            ⁡      t      −      1        t    )  (      0    0    f  o  r  m  )Applying L&#039;Hospital rule  L  =      lim          t      →      0        (                    e        t            (              cos        2            ⁡      t      −      sin      ⁡      2      t      )        1    )L=1  ⇒      lim          t      →      0        (  f  (  t  )      )          1              /            t        =  e

Let f be a differentiable function on R satisfying f'(t)=e^t(cos^2t-sin 2t) and f(0)=1, then which of the following is/are correct?

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New Questions in Calculus 2