Determine the critical points for each function. Determine...

Determine the critical points for each function. Determine whether the critical point is a local maximum or minimum, and whether or not the tangent is parallel to the horizontal axis. a. $h(x)=-6 x^{3}+18 x^{2}+3 \quad$ c. $y=(x-5)^{\frac{1}{5}}$ b. $g(t)=t^{5}+t^{3}$ d. $f(x)=\left(x^{2}-1\right)^{\frac{1}{3}}$

Determine the critical points for each function. Determine whether the critical point is a local maximum or minimum, and whether or not the tangent is parallel to the horizontal axis.
a. $h(x)=-6 x^{3}+18 x^{2}+3 \quad$ c. $y=(x-5)^{\frac{1}{5}}$
b. $g(t)=t^{5}+t^{3}$
d. $f(x)=\left(x^{2}-1\right)^{\frac{1}{3}}$

Key Concepts

Local Maximum and Minimum

Local extrema are the points where a function reaches a relative highest or lowest value compared to nearby points. Determining these involves analyzing the sign change of the first derivative around the critical points or using the second derivative test when applicable.

Horizontal Tangents

A horizontal tangent occurs at a point where the derivative is zero, indicating that the slope of the function at that point is flat. This is often associated with local maximum or minimum points, though not all points with a horizontal tangent are necessarily local extrema.

Critical Points

A critical point of a function is a point in its domain where the derivative is zero or does not exist. These points are significant because they are potential candidates where the function's behavior changes, such as shifting from increasing to decreasing, thus possibly indicating a local maximum or minimum.

First Derivative

The first derivative of a function represents its instantaneous rate of change. Setting the first derivative equal to zero is a common method for finding horizontal tangents and potential extrema, as these are the points where the slope of the function is zero.

Non-Differentiability

Non-differentiability refers to points in the domain of a function where the derivative does not exist, even if the function is continuous there. Such points need to be considered as critical points since the failure of the derivative to exist may coincide with notable changes in the function's behavior.

Transcript

00:01 For this problem, we want to find the critical points of a few functions, determine whether or not, whether they are max, min, or neither, and determine whether or not the tangent line will be parallel to the horizontal axis at that critical point.

00:14 So our first function is negative 6x cubed plus 18x squared plus 3.

00:25 And that is labeled as for this problem.

00:29 One moment, that is labeled as h of x.

00:34 So, taking h prime of x, we will get just 36x minus 18x squared.

00:49 Having that, we have a common factor of, in fact, it's a common factor of 18x between the two terms.

01:02 So we can actually factor out a negative 18x and write this as negative 18x times x minus 2, which tells us that we'll have a critical point at x equals 0 and a critical point at x equals 2.

01:16 Now, since we have a negative leading coefficient on an odd polynomial, that means that we'll essentially have double negatives when x is less than 0.

01:26 So we'll be coming in positive to the left of x equals 0, then turning to negative in between 0 and 2, then turning to positive again.

01:34 Now, that means that we'll have, oh, excuse me, it should be careful here.

01:42 We have on our derivative, we have a negative leading coefficient on an even even degree polynomial.

01:54 And so that actually means that we'll be coming in negative, turning to positive, then turning to negative again.

02:02 So what that means, going from negative to positive means that we have a min, and going from positive to negative, means that we have a max.

02:16 For the second part of the problem, part b, we have g of t equals t to the power of 5 plus tq.

02:29 It means that g prime of t is going to equal 5 t to the power of 4 plus 3t squared.

02:36 Can factor out the t squared from the two.

02:42 We have t squared times 5 t squared plus three.

02:45 This will only ever equal zero when t equals zero.

02:50 We have a critical point where t equals zero...

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