The Mean Value Theorem &nbs...

Question

The Mean Value Theorem

a. Show that the equation 𝓍⁴ + 2𝓍² ― 2 = 0 has exactly one solution on [0,1] .

[Technology Exercises] b.Find the solution to as many decimal places as you can.

Accepted Answer

Below there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the validity of the given statement. The equation 4x the power of 3 + 3X + 3 equals 0 has exactly one solution on square bracket -1.0 square bracket. Awesome. So it appears for this particular problem, we're asked to determine the validity, whether this statement that is given to us is true or false. So we're given an equation, and ultimately we're trying to figure out if this statement, based on the equation, and we're told that it has exactly one solution on this closed interval, that's what square brackets mean. Has one solution. So is that true or false? So our first multiple choice answer A is true and B is false. So we're trying to figure out if this statement, the equation 4X is the power 3 + 3 X + 3 equals 0, has exactly one solution in this closed interval -1.0. So with that said, first off, we need to identify if the equation 4 X of the power 3 + 3 X + 3 equals 0 has exactly one solution on our closed interval of -1.0, and in order to do this, we can recall and apply the intermediate value theorem and the mean value theorem. So, first off, let us verify that the function, and let's say that our function is F of X is equal to 4 X to the power of 3 + 3 X + 3 is continuous and differentiable on this closed interval of -1.0. And since F of X is a polynomial, it will be continuous and differentiable everywhere. And once again, because F of X is a polynomial function, it will be continuous and differentiable everywhere. So now our next step is we need to check the value of F of X at its end points of the interval, meaning we need to take F of -1. So we need to plug in a value of -1 in for X, so we need to take 4 multiplied by -1, the power 3 + 3 multiplied by -1 + 3, which when we plug in that expression into our calculator. You'll get an answer of -4, and similarly, if we plug in a value of 0, I'm not gonna rewrite the expression out again, but if we plug in a value of 0 in for X, we will find that F is 0 is equal to 3. So since F of -1 is equal to 4 and F of 0 is equal to 3 by the intermediate value theorem, and because the function changes a sign, there must be at least one root R within our closed interval, -1.0, such that F of R is equal to 0. So with that said, we need to find F of X. So F of X, which is the first derivative of our function of F of X, and this will be equal to D by DX of 4 X 3 + 3 X + 3, which is going to be equal to 12 x 2 + 3. Fantastic. So now our next step is we need to prove that R is a unique solution of the equation, assuming that there are at least two solutions A and B, within our given interval, such that F of A is equal to F of B, which is equal to 0. So with that said, as we should recall a note, the mean value theorem states that for a function F that is continuous on a closed interval, which note that our closed interval means square brackets A, B, Awesome. And is differentiable on the open interval parentheses Abs. That will mean that there exists at least 1. C, which is an element of A B in parentheses, such that F of C is going to be equal to F. Of B minus F of A. All divided by B minus A. So with that said, We now need to note that by the mean value theorem, we have some value C such that A is less than C and C is less than B, such that F of C is equal to 0. However, F of X is equal to 12 x 2 + 3, which is greater than or equal to 3 for all values of X. So, that will mean that it's a contradiction. Thus, we can go ahead and state that F of X is equal to 4 X the power of 3 + 3 X + 3 has a unique solution on square bracket 1.02, which is the close interval -1.0. So, moreover, since F of X is equal to 12 x of the power 2 + 3, which is greater than or equal to 0 on our closed interval, -1.0, this will mean that F of X is strictly increasing on the given interval. So therefore, without Dot since F of X is strictly increasing and the changes signed from negative to positive within our interval, this implies that there is exactly one route in our closed interval of -1.0, so that will mean our final answer without a doubt is true. So looking at our multiple choice answers, our final answer has to be multiple choice answer letter A, which once again is true. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

The Mean Value Theorem

a. Show that the equation 𝓍⁴ + 2𝓍² ― 2 = 0 has exactly one solution on [0,1] .

[Technology Exercises] b.Find the solution to as many decimal places as you can.

Key Concepts

Mean Value Theorem

Intermediate Value Theorem

Finding Roots of Polynomials

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