Suppose that f(x) = 2xsqrt2x^2 + 3 (A) Find all critical...

Suppose that f(x) = 2xsqrt{2x^2 + 3} (A) Find all critical values of f. If there are no critical values, enter None . If there are more than one, enter them separated by commas. Critical value(s) = (B) Use interval notation to indicate where f(x) is increasing. Increasing: (C) Use interval notation to indicate where f(x) is decreasing. Decreasing: (D) Find the x-coordinates of all local maxima of f. If there are no local maxima, enter None . If there are more than one, enter them separated by commas. Local maxima at x = (E) Find the x-coordinates of all local minima of f. If there are no local minima, enter None . If there are more than one, enter them separated by commas. Local minima at x = (F) Use interval notation to indicate where f(x) is concave up. Concave up: (G) Use interval notation to indicate where f(x) is concave down. Concave down: (H) Find all inflection points of f. If there are no inflection points, enter None . If there are more than one, enter them separated by commas. Inflection point(s) at x = (I) Use all of the preceding information to sketch a graph of f. When you're finished, enter a 1 in the box below. Graph Complete:

Transcript

00:02 Okay, so we have the function 2x square root of 2x squared plus 3.

00:19 Okay, we need to calculate all the critical points and inflection points and all that.

00:26 Okay, so for these problems you always need to take two derivatives.

00:29 So let's first take the first derivative.

00:31 So we have to use product rule.

00:33 So we first take derivative of 2x which is 2 times 2 square root square plus 3, and then we take a derivative of the square root function.

00:45 So we have 2x and then we take a derivative of the square root.

00:50 So we get square root of 2x squared plus 1 over square root of 2x squared and then times one half.

01:00 But then you have to remember you have to use chain rule on the inside portion.

01:03 So this the derivative of 2x squared plus 3 is going to be 4x.

01:10 Okay, and now let's simplify this a bit.

01:20 Okay, so now let's put it into common denominator.

01:24 So let's simplify this by putting it in a common denominator.

01:29 So let's do square root of 2x squared plus 3, and then we get 2 times 2x squared plus 3 plus 2x.

01:45 So this is going to be 2x squared, 4x squared.

01:49 Okay, so this is going to be 8x squared plus 3 over square root of 2x squared plus 3.

02:11 Okay, and then we need to also find the inflection points and all that.

02:14 So we also need to take second derivative.

02:20 So to do this, you need to use quotient rule.

02:23 So first we take derivative of the numerator, which is going to be 16x times the stuff in the denominator.

02:35 And then we take a derivative of the denominator, which is again, so we have to use the chain rule.

02:53 So we get again, derivative of denominator is going to be this.

02:56 And then we need to multiply by the numerator plus 3.

03:01 And then we divide by the denominator squared.

03:11 Okay, so the denominator squared is going to be going to be this.

03:26 Okay, and now let's try to simplify this a bit.

03:31 So again, we divide here to get 2x.

03:37 And then let's simplify the numerator by putting under common denominators.

03:46 So this is going to be 16x times 2x squared plus 3 plus 2x times 8x squared plus 3, all over square root of 2x squared plus 3.

04:14 And then we divide by 2x squared plus 3.

04:21 Okay, so this is going to be, so it's going to be 32x cubed plus 16x cubed, so it's going to be 48x cubed.

04:36 And then we get 48x plus 6x, that'll be plus 54x.

04:47 And then all this thing is going to be divided by 2x squared plus 3 to the 3 halves.

04:59 Okay, so that was a long calculation, but these are derivatives.

05:07 Okay, so now we have to check, find the critical point.

05:16 This is where f prime is equal to zero or undefined.

05:23 So this basically means when is the numerator equals zero and when is the denominator equal to zero? well, for f prime, if you look at this, the denominator is never zero, because 2x squared plus 3 is always strictly positive, never zero, right, it's always bigger than 3, or square root of 3.

05:52 So f can, f prime can never be undefined.

05:55 So the only place where it can equal zero is if you have 8x squared plus 3 equal to zero...

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