Maximum value of (arcsinx)^2 - 2arcsinx

f(x) has a global maximum at x = -1

f(x) has a global minimum at x = sin(1)

Step-by-step explanation:

Find and classify the global extrema of the following function:

f(x) = sin^(-1)(x)^2 - 2 sin^(-1)(x)

Find the critical points of f(x):

Compute the critical points of sin^(-1)(x)^2 - 2 sin^(-1)(x)

The domain of f(x) = sin^(-1)(x)^2 - 2 sin^(-1)(x) is {x element R : -1<=x<=1}

The interior of the domain is {x element R : -1<x<1}:

f(x) = sin^(-1)(x)^2 - 2 sin^(-1)(x) when -1<x<1

To find all critical points, first compute f'(x):

d/( dx)(sin^(-1)(x)^2 - 2 sin^(-1)(x)) = (2 sin^(-1)(x))/sqrt(1 - x^2) - 2/sqrt(1 - x^2):

f'(x) = (2 sin^(-1)(x))/sqrt(1 - x^2) - 2/sqrt(1 - x^2)

Solving (2 sin^(-1)(x))/sqrt(1 - x^2) - 2/sqrt(1 - x^2) = 0 yields x = sin(1):

x = sin(1)

f'(x) exists for all x such that -1<x<1:

(2 sin^(-1)(x))/sqrt(1 - x^2) - 2/sqrt(1 - x^2) exists for all x such that -1<x<1

The only critical point of sin^(-1)(x)^2 - 2 sin^(-1)(x) is at x = sin(1):

x = sin(1)

The domain of sin^(-1)(x)^2 - 2 sin^(-1)(x) is {x element R : -1<=x<=1}:

The endpoints of {x element R : -1<=x<=1} are x = -1 and 1

Evaluate sin^(-1)(x)^2 - 2 sin^(-1)(x) at x = -1, sin(1) and 1:

x | f(x)

-1 | 5.60899

sin(1) | -1

1 | -0.674192

The largest value corresponds to a global maximum, and the smallest value corresponds to a global minimum:

x | f(x) | extrema type

-1 | 5.60899 | global max

sin(1) | -1 | global min

1 | -0.674192 | neither

f(x) = sin^(-1)(x)^2 - 2 sin^(-1)(x) has one global minimum and one global maximum:

f(x) has a global maximum at x = -1

f(x) has a global minimum at x = sin(1)

answer: -2+√x,-2-√x

i hope this , i don't know if you're looking for a pair or not.

step-by-step explanation:

if you want to solve percent error follow these steps.

1. subtract the accepted value from the experimental value.

2. take the absolute value of step 1.

3. divide that answer by the accepted value.

4. multiply that answer by 100 and add the % symbol to express the answer as a percentage.

if you want to solve percent of change just do this.

1. work out the difference (increase) between the two numbers you are comparing.

2. divide the increase by the original number and multiply the answer by 100. if your answer is a negative number then this is a percentage decrease.

hope this : )