Find the critical point and the interval on which the given function is increasing or decreasing, and apply the First Derivative Test to the critical point. Let

Question

f(x)=4/x^2+1

Critical Point =
Is f a maximum or minumum at the critical point?

The interval on the left of the critical point is:
On this interval, f is_______ while f′ is ______

The interval on the right of the critical point is:
On this interval, f is_______while f′ is__________

Answer 1

What, no ideas on any of your homework dump? I'll do this one, and the steps should make it easy for you to try the others and show what you did.

As you surely recall,
f(x) is increasing where f'(x) > 0
f(x) is decreasing where f'(x) < 0
f(x) is a max or min where f'(x) = 0
So, for your function, assuming the usual carelessness with parentheses, is
f(x) = 4/(x^2+1)
f'(x) = -8x/(x^2+1)^2
Now, the denominator is always positive, so
f'(x) > 0 where x < 0
f'(x) < 0 where x > 0
f'(x) = 0 where x = 0

so,
f(x) is increasing on (-∞,0)
f(x) is decreasing on (0,∞)
f(0) is a maximum, since the slope changes from + on the left to - on the right

there are many good online graphing sites. plot your function there, and see that the above is correct