Find the critical points and inflection points. Determine...

Find the critical points and inflection points. Determine the local extrema and the intervals where the function is concave upward and concave downward. (a) f(x) = 4x e^{3x} (b) f(x) = x - sin(x), x ? [-?/2, 3?/2]

Transcript

00:01 Hi there, in this question we are given two functions.

00:04 In part a, we have the function f of x is equal to 4x times e to the power of 3x.

00:13 So what we need to find? we need to find the critical points and the inflection points.

00:17 So for the critical point, we need to just take the first derivative of the function and set the equation equal to 0.

00:27 So the first derivative of the function, which is 4 parentheses.

00:31 So two values are in terms of x so i'm going to just use the multiplication rule for the derivation the derivative of the x which is one times e to the power 3x and plus the derivative of next which is 3 times a to power 3x times x so this is the equation that we have for the first derivative and set this equation equal to 0 so e to the power 3x is the common factor so we have 1 plus 3x here so from here e to the power 3x cannot be 0.

01:04 That means 1 plus 3x is equal to 0.

01:07 That means x is equal to negative 1 over 3.

01:11 So we have the critical point here.

01:14 And what about the external points? so i'm going to just do the sign table for the first derivative, which is negative 1 over 3.

01:22 So the leading option for the first derivative, which is positive, positive and negative.

01:28 So that means the function is decreasing and then increasing.

01:31 So at x is equal to negative 3, we have the local minimum.

01:42 And so, and for the inflection point, for the inflection point, we need to just take the second derivative of the function and set equal to 0.

01:56 So we have already known the first derivative of the function is four times e to the power 3x plus 3 times e to the power 3x times x let's take the derivative of this function one more which is 4 parentheses 3 e to the power 3x plus 3 parentheses so let me just use 3 as the factor so 3 times e to the power 3x and plus 3 times the derivative of e to the power 3x 3 times e to the power 3x times x and set this equation equal to 0.

02:33 So the e to the power 3x, i mean 3 times e to the power 3x is the common factor and also there's one more multiplication for here.

02:43 So if i just take the parentheses, we got 1 plus 1, which is 2 plus 3x that is equal to 0...

Question

Find the critical points and inflection points. Determine the local extrema and the intervals where the function is concave upward and concave downward. (a) f(x) = 4x e^{3x} (b) f(x) = x - sin(x), x ? [-?/2, 3?/2]

Please give Ace some feedback