For each function, find (a) the critical numbers; (b) the open intervals where the function is i

For each function, find (a) the critical numbers; (b) the...

Key Concepts

Critical Numbers

Critical numbers of a function are the x-values at which the derivative is zero or fails to exist. They are the candidates for local extrema (maximums or minimums) and play a key role in analyzing the function's behavior with respect to increasing or decreasing trends.

Monotonicity (Increasing and Decreasing Intervals)

Monotonicity refers to the intervals where a function is consistently increasing or decreasing. By determining the sign of the first derivative, one can find where the function rises (derivative > 0) or falls (derivative < 0), which is essential for understanding its overall behavior and identifying relative extrema.

First Derivative Test

The first derivative test is a method used to analyze the behavior of a function around its critical numbers. It involves checking the sign changes of the derivative before and after a critical point to ascertain whether the function attains a local maximum, minimum, or neither at that point.

Product Rule

The product rule is a differentiation technique used when a function is the product of two or more functions. It states that the derivative of such a product is the derivative of the first function multiplied by the second, plus the first function multiplied by the derivative of the second. This rule is essential when differentiating functions created by combining polynomial and non-polynomial components.

Chain Rule

The chain rule is a fundamental differentiation rule used for composite functions. It assists in computing the derivative of a function that involves an inner function nested inside an outer function. In cases where exponential functions have exponents that are functions of x, the chain rule enables the correct calculation of derivatives.

Exponential Functions and Their Derivatives

Exponential functions are characterized by constant multiplicative growth or decay rates, and their derivatives are proportional to the function itself. The differentiation process often involves multiplying by the natural logarithm of the base, a property that is useful when dealing with functions that combine exponential and other functional forms.

Transcript

00:01 Hello, we're looking at chapter 5, section 1, problem number 34, given the function f of x equals x times 2 to the negative x squared, you want to find the critical numbers and the open intervals where the function f of x is increasing and or decreasing.

00:22 So we're going to start by finding the derivative of f because critical number.

00:31 Numbers occur when the first derivative is equal to zero or undefined.

00:36 So to do so i'm going to remember that d dx of 2 to the u is equal to 2 to the u times the natural log of our base, so natural log of 2, and then of course times u prime for chain rule.

01:00 So looking at my f of x, i see it as a product rule first and foremost, which means i'm going to go first times the derivative of the second plus the second times the derivative of the second.

01:13 So first x times the derivative of the second.

01:18 So derivative of 2 to the u again is 2 to the u, which is negative x squared, times natural log of the base, so natural log of 2 times the derivative of u, so derivative of negative x squared is negative to x.

01:39 So so far i have the first times the derivative of the second, and then plus the second, so 2 to the negative x squared, times the derivative of the first, derivative of x is, of course, 1.

01:56 So i want to clean this up a bit, recognizing i can factor out base 2 to the negative x squared from both terms.

02:07 And so when i do, i will be left with x times negative 2x, so negative 2x squared times the natural log of 2 to give me the first term.

02:19 And the second is just a plus 1.

02:24 So i can write that as 1 minus 2x squared natural log of 2, all over 2 to the positive x squared.

02:40 So i know to find for part a, to find my critical numbers, i need to set the derivative.

02:48 So f prime of x equal to zero or undefined.

02:51 It's not going to be undefined because when i set the denominator equal to zero, 2 to the x squared is always positive.

02:59 So another 0.

03:00 So i'm not going to get anything from that.

03:02 So then i'm just going to find my critical numbers by setting 1 minus 2x squared natural log of 2.

03:10 Equal to zero.

03:12 And so i'm going to wind up with, let's see, i'll move this over and then divide.

03:19 So i can write that as 1 equal to 2 x squared natural log of 2.

03:29 So x squared is going to equal.

03:33 If i move this, divide by 2 and natural log of 2.

03:40 And so when i take the square root, i'm going to get x equals so negative the square root of 1 over 2 natural log of 2 or x equals the positive square root of 1 over 2 natural log of 2...

Please give Ace some feedback