# Calculus

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!

1. 👍 0
2. 👎 0
3. 👁 481
1. I will assume you meant:
f(x) = (x^2 + 1)/(x^2 - 9)
= (x^2 + 1)/((x-3)(x+3))

a) for y intercepts , let x = 0
then y = 1/-9 = - 1/9
for x intercepts, let y = 0
(x^2 + 1)/(x^2 - 9) = 0
then x^2 + 1 = 0
x^2 = -1 , which has no real solutions
so there are no x-intercepts

b) using the quotient rule
dy/dx = ( (x^2-9)(2x) - (x^2+1)(2x))/(x^-9)^2
= -20x/(x^2 - 9)^2

c) --- critical values are found in many of the other parts.

d) for local extrema, dy/dx = 0
-20x/(x^2 - 9)^2 = 0
-20x = 0
x = 0 , then f(0) = -1/9

time to look at the graph using Wolfram, ( my other answers are confirmed here)
http://www.wolframalpha.com/input/?i=y+%3D+%28x%5E2+%2B+1%29%2F%28x%5E2+-+9%29

there is a local maximum at -1/9 , when x = 0

e) y '' = ((x^2 - 9)^2 (-20) + 20x(2)(x^2 - 9)(2x) )/(x^2 - 9)^4
= 60(x^2 + 3)/(x^2 - 9)^3

f) concave up or concave down depends on y''
if y'' is > 0 for a given x, then the curve is concave up at that value
if y'' is < 0 for .... concave down
While the numerator of y’’ is always positive,
the denominator of y’’ could be + or –
for -3 < x < 3 , the value of (x^2 – 9)^3 is negative, all all values yield a positive result.
So the curve is concave down for -3 < x < 3
For all other values it is concave up
Of course it is undefined for x = ± 3, giving us vertical asymptotes at
X = 3 and x = -3

g)
points of inflection occur when y'' = 0
60(x^2 + 3)/(x^2 - 9)^3 = 0
60(x^2 + 3) = 0
x^2 + 3 = 0
x^2 = -3
No solution, thus no point of inflection

1. 👍 0
2. 👎 0

## Similar Questions

1. ### Functions - math

The function f is such that f(x) = 2x + 3 for x ≥ 0. The function g is such that g(x)= ax^2 + b for x ≤ q, where a, b and q are constants. The function fg is such that fg(x)= 6x^2 − 21 for x ≤ q. i)Find the values of a

2. ### math

The cost of producing q articles is given by the function C=f(q)=100+2q. (A) Find the formula for the inverse function. (B) Explain in practical terms what the inverse function tells you. I am pretty sure the answer to A is

3. ### Calculus. I need help!

Write the composite function in the form f(g(x)). [Identify the inner function u=g(x) and the outer function y=f(u).] Then find the derivative dy/dx. y=√sinx √ is square root.

4. ### Math22

1. Write the composite function in the form f(g(x)). [Identify the inner function u = g(x) and the outer function y = f(u).] y= (g(x), f(u)) = 2. Write the composite function in the form f(g(x)). [Identify the inner function u =

1. ### algebra

2. The function f is defined as follows f(x)={4+2x if x0 a) Find the domain of the function b) Locate any intercepts c) Graph function d) Based on graph find range e) Is f continuous on its domain? a) The domain of the function f

2. ### inverse

If f(x)=cosx + 3 how do I find f inverse(1)? Thanks y = cos(x) + 3 the inverse of this is x = cos(y) + 3 solve for y and you have your inverse The cos function only has a range of [-1,1], so the range of f(x) is [2,4]. this means

3. ### Math (Limits)

Consider the following function. f(x) = 2x^2 − 8x Find the limit. lim (Δx->0) (f(x+Δx)-f(x))/(Δx) How would I set up the equation to work with that function? I know how to plug in the values usually, but how would I do this

4. ### calc

let f be function given by f(x)= Ln(x)/x for all x> 0. the dervative of f is given by f'(x)= (1 - Ln(x))/x squared. a) write equation for the line tangent to the graph of f at x=e squared b) Find the x-coordinate of the critical

1. ### Math Demand Functions

Given the function q=D(x)=k/x^n where k is a positive constant and n is an integer greater than 0. How would you find the elasticity of the demand function After you find the der. what do you do?

2. ### MATH

We now have d dx [x5 + y8] = 5x4 + 8y7y' = d dx [9] = 0. Rearranging this, we get 8y7y' = ?????????? ------------------------------------- A table of values for f, g, f ', and g' is given. x f(x) g(x) f '(x) g'(x) 1 3 2 4 6 2 1 8

3. ### math

Consider the following. lim x→−2 2x^2 − 2x − 12/x + 2 Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.) Write a simpler function that agrees with the given function at all but one

4. ### calculus

1. Which of the following expressions is the definition of the derivative of f(x) = cos(x) at the point (2, cos(2))? 2. Find the derivative of f(x) = |x + 2| at the point (1, 3) 3. Find f '(x) for f(x) = -2x3 + 3x2 - x + 15. 4.