Find the rate of change of main dependent variableWe have f : R → R , f ( x ) = x 2 + x sin ⁡ ( x ), and we need to find intervals of monotonicity. Here is all my steps: f ′ ( x ) = 2 x + x cos ⁡ ( x ) + sin ⁡ ( x ) f ′ ( x ) = 0 ⇒ x = 0 the only solution.Now I need to find where f′ is positive and negative. I don't want to put value for f′ to find the sign, I want another method. So I tried to differentiate the function again to see if f′ is increasing or decreasing: f ″ ( x ) = 2 − x sin ⁡ ( x ) + cos ⁡ ( x ), but I don't know if f ″ ( x ) ≥ 0 or f ″ ( x ) ≤ 0.How can I find the sign for f′ to determine monotony of f ?

Question

Find the rate of change of main dependent variableWe have   f  :      R    →      R    ,    f  (  x  )  =      x    2    +  x  sin  ⁡  (  x  ), and we need to find intervals of monotonicity. Here is all my steps:      f    ′    (  x  )  =  2  x  +  x  cos  ⁡  (  x  )  +  sin  ⁡  (  x  )      f    ′    (  x  )  =  0  ⇒  x  =  0 the only solution.Now I need to find where f′ is positive and negative. I don&#039;t want to put value for f′ to find the sign, I want another method. So I tried to differentiate the function again to see if f′ is increasing or decreasing:      f    ″    (  x  )  =  2  −  x  sin  ⁡  (  x  )  +  cos  ⁡  (  x  ), but I don&#039;t know if       f    ″    (  x  )  ≥  0 or       f    ″    (  x  )  ≤  0.How can I find the sign for f′ to determine monotony of f ?

Kelton Glover · Accepted Answer

Step 1You know that the function f′(x) is a continuous function with only one root. This means that the sign of the function is the same on   [  0  ,  ∞  ), and this means that no matter what value of   x  &amp;gt;  0 you take, you will find the sign of f′(x) on this interval.In some cases, it&#039;s good to take a particular value since that makes it easier. In your case, it&#039;s very easy to calculate       f    ′    (  2  π  )  ,, for example.Also, you can try to see what the limit of f′(x) is when x becomes large. If the limit is positive (or if it is   ∞), then the sign of f′ is also positive on   [  0  ,  ∞  ). In your case, the limit is easy to calculate.

We have f:R rightarrow R,f(x)=x^2+x sin (x), and we need to find intervals of monotonicity.

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