Question

Find the intervals of increase and decrease of 1/x^2-9
b) find the coordinates of the local max and min
Please help, I got x=0 and x=+-3 but im not sure what the intervals would be for the increase and decrease

Answer 1

the first derivative of your function is:
\[\Large f'\left( x \right) = \frac{{ - 2x}}{{{{\left( {{x^2} - 9} \right)}^2}}}\]

Answer 2

am I right?

Answer 3

yes

Answer 4

ok! Now we have to establish where that first derivative is positive

Answer 5

So we have to solve this inequality:
\[\Large \frac{{ - 2x}}{{{{\left( {{x^2} - 9} \right)}^2}}} > 0\]

Answer 6

what are the solutions of that inequality?

Answer 7

would you set the denominator to = 0? i got +-3

Answer 8

no, since the denominator is always positive, then that fraction will be positive, if and only if also the numerator will be positive, namely:
\[\Large - 2x > 0\]

Answer 9

the denominator:
\[\Large {{{\left( {{x^2} - 9} \right)}^2}}\]
is a square, which is always positive, and it is equal to zero at x=3 and x=-3
That means our first derivative doesn't exist at x=3 and x=-3.
Also your function doesn't exist at x=3 and x=-3 for the same reason

Answer 10

Oh okay that makes sense

Answer 11

ok! Now what are the solutions of:
\[\Large - 2x > 0\]

Answer 12

it would be 0

Answer 13

if I multiply, both sides of that inequality by -1, I get:
\[\Large 2x < 0\]
am I right?

Answer 14

yes

Answer 15

ok! Now please, divide both sides of the last inequality by 2, what do you get?

Answer 16

x<0 (0/2)

Answer 17

perfect!

Answer 18

that means our function is an incresing function when x<0 and a decreasing function when x>0:

Answer 19

increasing*

Answer 20

so what can you conclude?

Answer 21

Okay, would my intervals be then increasing from (-infinity,0) and decreasingfrom (0,-infinity)?

Answer 22

that's right! Please keep in mind that at subsequent points:
x=-3 and x=3, our function is not defined

Answer 23

your last statement means that x=0 is a point of local maximum for f(x), am I right?

Answer 24

decreasing in (0, +infinity) and increasing in (-infinity, 0)

Answer 25

thats what i have, but Im not sure if its right, would the -3 and 3 have anything to do with the local max and min?

Answer 26

no, they are not. Those points don't belong to the domain of our function