Intervals of Increase and Decrease Explained: In-Depth Examples, Equations, and Expert Insights

Recent questions in Intervals of Increase and Decrease
Integral CalculusAnswered question
Moselq8 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?

Intervals of increase and decrease are important in calculus and other mathematical disciplines. They can be used to find extreme values of functions and to understand the behavior of functions. There are several ways to calculate intervals of increase and decrease. The most common is to take the derivative of the function and then find the points where the derivative is zero or undefined. They also can be found by graphing the function and then looking at the points where the graph changes from concave up to concave down. If you have any questions about intervals of increase and decrease, feel free to ask our a math tutors or teachers for help.