Find the general solution of the given system.

dx/dt=6x-y

dy/dt=5x+4y

“Find the critical values at which each of the following cubic functions is optimized and test the second-order conditions to see if the function is at a relative maximum, relative minimum, inflection point, or saddle point.

“(c) z = 8x^2 − 6y^3 + 144xy − 323
(d) z = 6x^3 + 6y^3 + 54xy − 195”

ans((c) (0, 0), inflection point; (648, −72), relative minimum
(d) (0, 0), inflection point; (−3, −3), relative maximum) show process

Initially 10 grams of salt are dissolved into 35 liters of water. Brine with concentration of salt 4 grams per liter is added at a rate of 5 liters per minute. The tank is well mixed and drained at 5 liters per minute.

A.Let x be the amount of salt, in grams, in the solution after t minutes have elapsed. Find a formula for the rate of change in the amount of salt, dx/dt, in terms of the amount of salt in the solution x.
dxdt= grams/minute

B. Find a formula for the amount of salt, in grams, after t minutes have elapsed.
x(t)= grams

C. How long must the process continue until there are exactly 20 grams of salt in the tank?

minutes

f(x) = 10/cos-1(x). Use calculus to determine:

a) all critical values

b) any local extrema

c) any absolute extrema

d) the intervals where f is increasing/decreasing

e) any points of inflection rounded to the thousandths place

f) intervals where f is concave up/down

No interval was specified so I assumed 0<x<2pi

also I didn't get any concavity not sure if I'm right or not.

I got x= pi as the critical value as well as the relative and absolute minimum extrema.

would greatly appreciate some help

1A) Let ?(?) = 3? + 2. Use the ? − ? definition to prove that lim?→1 3? + 2 ≠ 1.

Definition and proof.

1B) Let ?(?) = 2?^2 − 4? + 5. Use the ? − ? definition to prove that lim?→−1 2?^2 − 4? + 5 ≠ 8.

definition and ? − ? Proof.

Consider the function below. (If an answer does not exist, enter DNE.)

h(x) = (x + 1)⁹ − 9x − 2

(a) Find the interval of increase. (Enter your answer using interval notation.)

Find the interval of decrease. (Enter your answer using interval notation.)

(b) Find the local minimum value(s). (Enter your answers as a comma-separated list.)

Find the local maximum value(s). (Enter your answers as a comma-separated list.)

(c) Find the inflection point.
(x, y) =

Find the interval where the graph is concave upward. (Enter your answer using interval notation.)

Find the interval where the graph is concave downward. (Enter your answer using interval notation.)

(d) Use the information from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one.

Find T(t), N(t), and B(t) for r(t) = t^2 i + (2/3)t^3 j + t k at the point P ( 1, (2/3) , 1)

Estimate the slope of the line from its graph.

The curve C is given by the parameterization ⃗r(t) = <−t , 1 − t^2> for −1 ≤ t ≤ 1.
a) Choose any vector field F⃗ (x, y) = 〈some function , some other function〉 and setup the work integral of F⃗ over C.

b)Choose any vector field G⃗(x,y) which has a potential function of the form φ(x,y)= x^3 + y^3 + some other stuff and compute the work done by G⃗ over C.

Please use a somewhat basic function (something other than <x^2 , y> please). For a thumbs up, please show all work and explain it clearly and write neatly. Thank you

1. Consider the cubic function f ( x ) = ax^3 + bx^2 + cx + d where a ≠ 0. Show that f can have zero, one, or two critical numbers and give an example of each case.

2. Use Rolle's Theorem to prove that if f ′ ( x ) = 0 for all xin an interval ( a , b ), then f is constant on ( a , b ).

3.True or False. The product of two increasing functions is increasing. Clarify your answer.

4. Find the point on the graph of f ( x ) = 4 − x 2 that is closest to the point ( 0 , 1 ).

Suppose that in a given store, customers are willing to buy y pounds of coffee beans per day at $x per pound, as given by the price-demand equation y=10+(180/x) for 2 (is less than or equal to) x (is less than or equal to) 10.

a. Find y’ with appropriate units. Explain what the first derivative means in this situation by hand.

b. Find the demand and the instantaneous rate of change of demand with respect to price when the price is $5.

c. Use a table of values to sketch a well-labelled graph of this function between $2 and $10 per pound.

d. Does the rate of demand increase or decrease over time? Explain.

3. Suppose a bird takes off from the top of a lamp post with its altitude (in m) given by

p(t) = (1/3)t³ - (1/2)t² -6t +15

(a) Calculate the bird’s velocity and acceleration.

(b) How tall is the lamp post?

(c) Is the bird initially flying up or down as it leaves the lamp post? Justify your answer using concepts from calculus.

(d) Determine the critical points for p(t).

(e) Determine the possible points of inflection for p(t).

(f) Determine the intervals on which p(t) is concave up and concave down.

Use spherical coordinates.

Evaluate

If x 1 and x 2 are the number of items of two goods bought, a customer’s utility is U ( x 1 , x 2 ) = 2 x 1 x 2 + 3 x 1 . The unit cost is $ 1 for the first good and $ 3 for the second. Use Lagrange multipliers to find the maximum value of U if the consumer’s disposable income is $ 100 . Estimate the change in optimal utility if the consumer's disposable income increases by $ 8 .