Calculus Answers

Apply the Young's theorem to justify that:

fxy(1,1)= fyx(1,1) for the function f : R^2→R , defined by

f(x,y)= |x+y|

find two positive numbers whose product is 64 and whose sum is a minimum

if the length of the hypotenuse of a right triangle is 10, find the lengths of the other sides when the area is a minimum

The function,f: [-1,1]×[2,1] → R, defined by

f(x,y)= { x ; y is rational

{ 0 ; y is irrationa

is integrable or not.

True or false with full explanation

Determine the area of the region bounded by the given set of curves. x=e^y, x=1, y=1 and y=2

L'Hospital rule.

(i) the interval(s) over which f rises and where it falls;

(ii) the local extrema.

Solve the following problems involving optimization.

1. A rectangular is to be fenced off along the bank river where no fence is required along the bank. If the material for the fence costs 60 pesos per running foot for the two ends and 90 pesos per running foot for the side parallel to the river, find the dimensions of the field of the largest possible area that can be enclosed with 27,000 pesos worth of fence.

2. Find a pair of non-negative number that have a product of 162 and minimize the sum of two times the first number and second number with closed bounded internal of [1,10].

1.f(x) = 4x^2 − 1/12x^2 + x − 1.Differentiate

Given that f"(x)=2-2/x3 and f'(1)=0,find f'(x).Given further that f(1)=8, find f(x).