Find the critical points of the following functions on the giv...

Question

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

g(x) = x sin⁻¹ x on [-1, 1]

Accepted Answer

Determine the critical points of F of X equals X, cosine inverse X on the interval -1 to 1. What are the absolute maximum and minimum values of F of X? Then we have 4 possible answers. We're looking for a critical point, absolute max, and an absolute mint. Now, to solve this, we first need to find the critical point. To find the critical point We will set our derivative F X equals to 0. So, we have F of X equals X, cosine inverse of X. And to find this derivative, we can make use of the product rule. This tells us if we have two functions multiplying each other. U and V We can write this as U multiplied by V. Plus you multiplied by V. In this case, U is X, V is cosine inverse. You would be 1. And the V is -1 divided by the square root of 1 minus X2. These are our known derivatives. Now, we can put this into the product rule. F X equals 1, multiplied by cosine inverse of X. Plus X multiplied by -1 divided by the square root of 1 minus X2. We get cosine inverse of X minus X divided by the square root of 1 minus X squared. Now, we'll set this equal to 0. And to solve this, we can place this in our calculator. We would end up getting X equals 0.65 when placing this into the graphic calculator. So, we have our critical point. We now need to find Our max and mint. To find our maximum and minimum value, let's plug in values of our function. To find our maximin. We have F of 0.65. It should be 0.65, cosine inverse of 0.65. And this is approximately 0.56. We can also plug in the values on our interval. F of -1. is -1 Cosine inverse of -1. Which gives us Negative pipe And if a positive one. Which is one Cosine in the verse of one. Which gives us 0. We can then see The max occurs when X equals 0.65. And their output It's 0.56. And our minimum Occurs when X equals -1. And the output Is negative pi. If we look at our possible answers, We see the answers to our problem. This answer B. OK, I hope to help you solve this problem. Thank you for watching. Goodbye.

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
g(x) = x sin⁻¹ x on [-1, 1]

Key Concepts

Critical Points

Absolute Maximum and Minimum

Inverse Sine Function

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