Absolute maxima and minima Determine the location and value of the absolute extreme values of f

step 1 Let's find the absolute max and mince for this function. If x equals x squared plus arc cosine o

Absolute maxima and minima Determine the location and value...

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as the inverse cosine function, are used to 'undo' the effect of trigonometric functions. Their inclusion in a function can introduce additional complexities in the computation of derivatives and the evaluation of the function.

Differentiation

Differentiation is the process used to compute the derivative of a function, which in turn is used to identify critical points where the rate of change is zero or undefined. This tool is central to analyzing and solving optimization problems.

Extreme Value Theorem

The Extreme Value Theorem guarantees that a continuous function on a closed interval must attain both an absolute maximum and an absolute minimum. This is a fundamental concept in calculus that provides assurance that the optimization problem over such an interval has a solution.

Critical Points

Critical points are values in the domain of the function where its derivative is zero or undefined. They are key candidates for local extrema and must be evaluated alongside the endpoints in the search for absolute extreme values.

Endpoint Evaluation

When searching for absolute extrema on a closed interval, it is essential to evaluate the function at the endpoints of the interval. This complements the search for extrema at critical points since the extreme values could occur at these boundary points.

Transcript

00:01 Let's find the absolute max and mince for this function.

00:04 If x equals x squared plus arc cosine on the interval from negative 1 to 1.

00:09 Just a reminder, arc cosine is the same thing as the inverse cosine written like this.

00:16 But i prefer to think of it as arc cosine because sometimes people make the mistake of seeing it like this where it's a power and not a inverse function.

00:26 Okay.

00:28 So to find absolute max and minns, we have to find critical value.

00:30 Or where the derivative is equal to zero.

00:34 So let's find the derivative.

00:37 So x squared goes to 2x, and then the derivative of arc cosine is negative 1 over the square root of 1 minus x squared.

00:47 And we want to know when this is equal to 0.

00:51 So if we bring the fraction to the other side and then multiply it over, we'll get 2x times the square root of 1 minus 0.

01:07 X squared is equal to one.

01:09 I'm going to square both sides.

01:13 So then you get four x squared times one minus x squared is still equal to one.

01:25 And then we can distribute to get four x squared minus four x to the fourth equals one.

01:35 Okay, now i'm going to bring everything to one side and rewrite it to get four x to the fourth.

01:41 Plus 4x squared minus 1 is equal to 0.

01:46 Now we have this x to the 4th, this x squared.

01:48 It looks kind of scary.

01:50 How do we solve this? but we can do it by doing a substitution.

01:53 We're going to let a equal x squared, so then we can have negative 4a squared plus 4a minus 1 equal to 0.

02:03 Now this is quadratic, so we're just going to go right for the quadratic equation.

02:06 So we'll get negative 4 plus or minus the square root.

02:10 4 squared minus 4 times negative 4 times negative 1 all over 2 times negative 4.

02:23 Well, let's see this is 16.

02:25 There's 3 negatives, so it's minus 4 times the 4 is 16 again.

02:30 So we'll have negative 4 plus or minus square root of 16 minus 16 over 8.

02:37 So then we get negative 4 over negative 8 or a half.

02:45 Now this is not the value for x.

02:46 This is the value for a.

02:50 Remember we said a equals x squared.

02:55 That was a substitution.

02:56 So if we like 1 1⁄2 equal x squared, then we can solve for x...

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