Math(Please check. Thank You)
posted by Hannah on .
1) solve:e^x(x^24)=0
e^x=0 and then (x+2)(x2) so x=2, 2
2) differentiate: y=ln(6x^2  3x + 1)
1/(6x^2  3x + 1) * 12x3
3) differentiate: y=e^3x+2
3 * e^3x+2
4) evaluate: 2^4x=8
2^4x = 2^3
4x = 3
x=1 so x=1
5) differentiate: x^3 + y^3 6 =0
3x^2 + 3y^2
6) A rectangular garden has an area of 100 square meters for which the amount of fencing needed to surround the garden should be as small as possible.
a) draw a picture of a rectangle and select appropriate letters for the dimensions
I chose x and y
b)determine the objective and constraint equations
objective: A=xy
constraint= 100=xy
c) find the optimal values for the dimensions.
I am not sure for this one
Thank you for your help!!!

1. e^x can never equal zero, but can approach it as x>>infinity
But the roots are in fact 2,2
2 correct
3 e^(3x+2)=e^3x * e^2
then y'= 3e^3x * e^2=3e^(3x+2)
4, correct
5. No.
x^3+y^3=0 You can do implicit differentation, but I am not certain you know that.
y^3=x^3
y= cubroot x^3=x
y'=1
6) xy=100 constraint
Perimeter= 2x+2y objective
dP/dx= 2+ d2(200/x)/dx= 2400/x^2 =0
2x^2=400
x=10 Y=10 is minimum fencing 
5) (x^3+y^36)'=3x^2+3y^2*y'=0
y'=x^2/y^2, where y=(6x^3)^(1/3) 
I agree, Mgraph, thanks.

Thank you!!