A function f(x) and interval [a, b] are given. Check if the...

A function $f(x)$ and interval $[a, b]$ are given. Check if the Mean Value Theorem can be applied to $f$ on $[a, b]$. If so, find all values $c$ in $[a, b]$ guaranteed by the Mean Value Theorem. $f(x) = 2x^3 - 9x^2 - 24x + 6$ on $[-6, 9]$ c = (Separate multiple answers by commas.) Note: You may need to use the quadratic formula for this problem. Recall that the solutions to $ax^2 + bx + c = 0$ are $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. If the Mean Value Theorem does not apply, enter DNE for the c value.

Transcript

00:01 So we want to find or rather use the mean value theorem.

00:05 We need to make sure a function is continuous on an interval that we're given, which this function is.

00:10 Ln is actually continuous on the interval from x greater than zero to x equals infinity.

00:25 So as long as x is greater than zero, the natural algorithm is continuous.

00:29 And of course, seven is continuous on any interval.

00:32 So that's checked.

00:33 Now it's differentiable on the interval.

00:34 So it is, there are no holes or discontinuities on this interval.

00:37 So we can take the derivative.

00:39 There are no cusps or corners either.

00:41 So we take the derivative.

00:42 And what the mean value theorem says, so the mean value theorem says that if we have those conditions met, then we can take ultimately what looks like the slope.

00:58 We can take the slope and this should be equal to the derivative at some point c in between the ending.

01:12 Now what this looks like, this is basically saying, okay, another change.

01:17 If i can get some values a and b, let's say a is here and b is here.

01:27 If i can get some values that are well, well founded enough on this interval, then measuring the derivatives along this interval would be very similar to measuring the slope of the tangent of the secant line on this interval.

01:49 Obviously the more linear your curve is, closer to the truth this will be.

01:55 So that's what we need to test.

01:56 We need to test that the derivative at some point, and we need to figure out if that point in the interval will give us the average slope, the average rate of change.

02:05 So first let's find the average rate of change.

02:07 F of b minus f of a over b minus a.

02:10 We know that a and b is one in 12.

02:12 So we have f of b, or sorry, f of 12 minus f of a, which would be one, over b minus a, which would be 12 minus one, is going to be equal to, that will be, well let's write it out.

02:33 This is going to be eight natural logarithm of 12 plus seven minus eight natural logarithm of one plus seven over 12 minus one is 11...

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