Sketch f ( x ) = sin ⁡ x + 1 x finding local maxima and minima, intervals of increase and decrease. I'm trying to use differentiation to draw this picture and find critical points.So, I get f ′ ( x ) = cos ⁡ x − 1 x 2 . However, I'll have to deal with inequality f ′ ( x ) &gt; 0, and f ′ ( x ) = 0, I feel I lack an ability to solve an equation like this. So is there any better way to find local maximun and minumum for this function?

Question

Sketch   f  (  x  )  =  sin  ⁡  x  +      1    x   finding local maxima and minima, intervals of increase and decrease. I&#039;m trying to use differentiation to draw this picture and find critical points.So, I get       f    ′    (  x  )  =  cos  ⁡  x  −      1          x      2      . However, I&#039;ll have to deal with inequality       f    ′    (  x  )  &amp;gt;  0, and       f    ′    (  x  )  =  0, I feel I lack an ability to solve an equation like this. So is there any better way to find local maximun and minumum for this function?

raffatoaq · Accepted Answer

Step 1For f′, note that, if       |    x      |    &amp;lt;  1, then       f    ′    (  x  )  ≠  0 (do you see why?).Also, for large x, since   cos  ⁡  (  x  )  =  sin  ⁡  (  π      /    2  −  x  ), you want   sin  ⁡  (  π      /    2  −  x  ) small. Therefore,   π      /    2  −  x has to be close to a multiple of   π, since that is when sin is small. So, let   π      /    2  −  x  =  π  n  +  z, where z is small. Then   cos  ⁡  (  x  )  =  sin  ⁡  (  π      /    2  −  x  )  =  sin  ⁡  (  π  n  +  z  )  ≈  (  −  1      )    n    z, so       1          x      2        ≈  (  −  1      )    n    z  ≈  (  −  1      )    n    (  π      /    2  −  π  n  )  −  x.Step 2Since x is large,   x  ≈  (  −  1      )    n    (  π      /    2  −  π  n  ).Plot f and f′ and see if these remarks agree with what you see.Remember, the most useful approximation for trig functions is   sin  ⁡  (  x  )  ≈  x for small x.

Sketch f(x)=sin x+1/x finding local maxima and minima, intervals of increase and decrease. I'm trying to use differentiation to draw this picture and find critical points.

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