Given the function: f(x) = x^2 + 1 / x^2 - 9

a)find y and x intercepts
b) find the first derivative
c) find any critical values
d) find any local(relative) extrema
e) find second derivative
f) discuss the concavity
g) find any inflection points

Please show me how u got this!

Sure! Let's go step by step to find the answers to these questions.

a) To find the y-intercept, we substitute x = 0 into the function and solve for y. So, we have:

f(0) = (0^2 + 1) / (0^2 - 9)
= 1 / (-9)
= -1/9

Therefore, the y-intercept is (0, -1/9).

To find the x-intercepts, we set the numerator (x^2 + 1) equal to zero and solve for x. So, we have:

x^2 + 1 = 0
x^2 = -1 (which has no real solutions)

Therefore, there are no x-intercepts for this function.

b) To find the first derivative, we differentiate the function with respect to x. Using the quotient rule, we have:

f'(x) = [(2x)(x^2 - 9) - (x^2 + 1)(2x)] / (x^2 - 9)^2

Simplifying this expression gives the first derivative.

c) To find the critical values, we set the first derivative equal to zero and solve for x. The critical values occur where the derivative is either zero or undefined. So, we solve the equation:

f'(x) = 0

d) To find the local (relative) extrema, we can use the critical values found in the previous step and apply the first or second derivative test. The first derivative test involves checking the sign changes in the derivative around the critical values to determine whether a local maximum or minimum occurs at that point.

e) To find the second derivative, we differentiate the first derivative with respect to x.

f''(x) = d^2/dx^2[ f'(x) ]

Simplifying this expression gives the second derivative.

f) To discuss the concavity, we can examine the sign changes of the second derivative and determine where the function is concave up or down.

g) To find the inflection points, we set the second derivative equal to zero and solve for x. The inflection points occur where the concavity changes.

Now that you know the steps involved, you can apply them to the given function f(x) = x^2 + 1 / x^2 - 9 to find the specific answers for each part.