SOLUTION: Find the intervals on which f is increasing and the intervals on which it is decreasing. f(x)= x^2 ln x^2+3

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Question 1155431: Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x)= x^2 ln x^2+3
Found 3 solutions by greenestamps, MathLover1, ikleyn:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


Clarify....

x^2 ln(x^2) + 3 = ?

or

x^2 ln(x^2+3) = ?
Answer by MathLover1(20849)   (Show Source): You can put this solution on YOUR website!

-> parabola, opens up
domain:
(,)
range:
{ element : } (all non-negative real numbers)
x-intercept:


=>x-intercept at origin

y-intercept:


=>y-intercept at origin

=> minimum is at origin
so, the interval on which is increasing is
(,)

and the interval on which it is decreasing is
(,)



Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.


    

    Plot y = (x^2)*(ln(x^2+3)



Looking into the formula, you can see that the function  f(x) = (x^2)*ln(x^2+3)  


    - first, is defined at all values of x (over all the domain of real numbers) and is even function,

    - and second, that it is MONOTONIC in the domain x >= 0.


Indeed, than larger the argument x is, than larger each of both factors  x^2  and ln(x^2+3) is.


So the function f(x) is monotonically increasing in the domain x >= 0.


Then from the fact that it is even function, you may conclude that the function is monotonically DECREASING in the domain  x < 0.


So you can perform all the necessary analysis without using Calculus, i.e., practically, MENTALLY.



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