Figuring out when f ( x ) = sin ⁡ ( x 2 ) is increasing and decreasingRegarding the function f ( x ) = sin ⁡ ( x 2 ), I'm supposed to figure out when it is increasing/decreasing.So far, I've found the derivative to be f ′ ( x ) = 2 x cos ⁡ ( x 2 ).So long as I can solve the inequality cos ⁡ ( x 2 ) &gt; 0. I can figure the rest out, but this is where I'm stuck.I've narrowed it down to − π 2 &lt; x 2 &lt; π 2 meaning x 2 &lt; π 2 ± n 2 π , n ∈ N .Then I get x &lt; ± π 2 ± n 2 π .Am I on the right track here? I can't seem to find a path from here.

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Figuring out when   f  (  x  )  =  sin  ⁡  (      x    2    ) is increasing and decreasingRegarding the function   f  (  x  )  =  sin  ⁡  (      x    2    ), I&#039;m supposed to figure out when it is increasing/decreasing.So far, I&#039;ve found the derivative to be       f    ′    (  x  )  =  2  x  cos  ⁡  (      x    2    ).So long as I can solve the inequality   cos  ⁡  (      x    2    )  &amp;gt;  0. I can figure the rest out, but this is where I&#039;m stuck.I&#039;ve narrowed it down to   −      π    2    &amp;lt;      x    2    &amp;lt;      π    2   meaning       x    2    &amp;lt;      π    2    ±  n  2  π  ,        n  ∈      N  .Then I get   x  &amp;lt;  ±            π      2        ±    n    2    π  .Am I on the right track here? I can&#039;t seem to find a path from here.

Addison Herman · Accepted Answer

Step 1You&#039;re on the right track.The &quot;points that matter&quot; in partitioning the increasing/decreasing intervals are the points for which       f    ′    (  x  )  =  0. This happens when either   2  x  =  0 or   cos  ⁡  (      x    2    )  =  0.The function is increasing when       f    ′    (  x  )  &amp;gt;  0, and decreasing when       f    ′    (  x  )  &amp;lt;  0.Consider each part of the derivative separately.The linear part of the derivative, 2x, is positive when   x  &amp;gt;  0, and negative when   x  &amp;lt;  0.Step 2The cosine function part of the derivative,   cos  ⁡  (      x    2    ), is positive when its argument is on one the open intervals  (  2  π  n  −  π      /    2  ,  2  π  n  +  π      /    2  )  ,  n  ∈      Z    ,and negative when it&#039;s on one the open intervals   (  2  π  n  +  π      /    2  ,  2  π  n  +  3  π      /    2  )  ,  n  ∈      Z    .The next steps would involve looking at when the product of these two parts changes sign. Those points would define your increasing/decreasing intervals.

Figuring out when f(x)=sin(x^2) is increasing and decreasing.

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