find critical points from such a freak equation?

Question

Find the critical point and state if the parameter effects the critical point. Identify if the critical point(s) are a maximum or a minimum

1. E(θ) = (θ-µθ^2)/ (µ+ θ) Assume µ is a positive constant and θ >
0
2. S = (7k/x^2) + k/(20-x)^2 Assume k is a positive constant

Answer 1

I may be able to help you a little. The first derivative comes out to [-u(theta ^2 + 2utheta-1)]/(u + theta)^2. Since u is a positive constant and theta is > 0. We can tell that the first derivative is going to be negative. The second derivative comes out to (-2utheta - 2u^2 + 1)/2(u + theta). Sorry I'm just now learning this so this is as far as I can go.

Answer 2

Get the first derivative. Equate it to zero. Find the parameter by rearranging the equation. Take the critical value as zero, and substitute in the equation for theta and see what the initial condition is. So if the parameter changes, the critical point shifts.
Use this rule for the second part of the question. The critical points are the initial conditions and the parameters are the roots. So the roots can be maxima or minima and not the initial conditions.

Answer 3

Allow me to switch notation so I can think on it easier

E(x) = (x - ax^2)/(a + x)= (x-ax^2)*(a + x)^-1

Using the the product rule the dervative is

(x - ax^2)* [-1(a+x)]^-2 + (1 -2ax)/(a+x)^-1

= -a[x^2 + 2a -1]/(a+x)^2

= -a[(x+a)^2 + (1-a^2)]/(a+x)^2

since both x and a are >o the denominator will not shrink to zero.

So, providing I made no errors, the numerator will equal zero when x = -a +/- sqrt((1+a)(1-a))

that is if no imaginary numbers do not crop up, if a is greater than 1 it will and there will be no critical pts.

Answer 4

a.dE/dtheta
=[(a+theta)(1-a)-(theta-atheta...
=[a+theta-a^2-atheta-theta+ath...
=[a(1-a)/(a+theta)^2]
b.dS/dx
=-2kX^-3-2k(20-x)^-3(-1)
=-2[k/x^3-1/(20-x)^3]