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Recent questions in Intervals of Increase and Decrease
Moselq8 2022-08-13

If h has positive derivative and φ is continuous and positive. Where is increasing and decreasing f
The problem goes specifically like this:
If h is differentiable and has positive derivative that pass through (0,0), and φ is continuous and positive. If:
f ( x ) = h ( 0 x 4 4 x 2 2 φ ( t ) d t ) .
Find the intervals where f is decreasing and increasing, maxima and minima.
My try was this:
The derivative of f is given by the chain rule:
f ( x ) = h ( 0 x 4 4 x 2 2 φ ( t ) d t ) φ ( x 4 4 x 2 2 )
We need to analyze where is positive and negative. So I solved the inequalities:
x 4 4 x 2 2 > 0 x 4 4 x 2 2 < 0
That gives: ( , 2 ) ( 2 , ) for the first case and ( 2 , 2 ) for the second one. Then (not sure of this part) h ( 0 x 4 4 x 2 2 φ ( t ) d t ) > 0 and φ ( x 4 4 x 2 2 ) > 0 if x ( , 2 ) ( 2 , ) . Also if both h′ and φ are negative the product is positive, that's for x ( 2 , 2 ).
The case of the product being negative implies:
x [ ( , 2 ) ( 2 , ) ] ( 2 , 2 ) = [ ( , 2 ) ( 2 , 2 ) ] [ ( 2 , ) ( 2 , 2 ) ] = .
So the function is increasing in ( , 2 ) , ( 2 , 2 ) , ( 2 , ). So the function does not have maximum or minimum. Not sure of this but what do you think?