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

step 1 Hello, we're looking at chapter 5, section 1, problem number 26. In a function f of x equals the

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

Key Concepts

Monotonicity Analysis

Monotonicity analysis involves determining the intervals over which a function is increasing or decreasing by examining the sign of its derivative. When the derivative is positive, the function is increasing, and when it is negative, the function is decreasing. This analysis is crucial in understanding the overall behavior of the function.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It allows one to differentiate a function that is composed of two or more functions by differentiating the outer function and then multiplying by the derivative of the inner function.

Critical Numbers

Critical numbers of a function are the points in the domain where its derivative is either zero or does not exist. They are significant because they can indicate potential local maxima, minima, or points of inflection, and are therefore key in analyzing the function's behavior.

Derivative and Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to a variable. In this context, correctly computing the derivative is essential to locate critical numbers and determine where the function is increasing or decreasing.

Transcript

00:01 Hello, we're looking at chapter 5, section 1, problem number 26.

00:06 In a function f of x equals the square root of x squared plus 6, and we're asked to find the critical numbers and intervals of increasing and decreasing.

00:19 So to start, i'm going to go ahead and rewrite my square root as x squared plus 6 to the 1 half power.

00:31 And i'm going to do that so that as to find my critical numbers, i need to find the first derivative and set it equal to zero or undefined.

00:40 And so i want to have it set up for a power rule.

00:44 And of course, it's going to include the chain rule.

00:48 So remember for the chain rule, what we're doing is we're going to say f prime of x is equal to.

00:53 So my outside function is something or whatever is in here.

00:59 Some teachers will call it u, so u to the one -half.

01:04 So we're going to use the power rule, put the one -half in front of whatever is here, and then subtract one.

01:11 So i'll write that as one -half, and then my parentheses, so that stays the same inside, x squared plus six, and we're going to subtract one, so get negative one -half.

01:23 But then for chain rule, we multiply times the derivative of the inside part, right? outside times inside.

01:33 So the derivative of x squared plus six is 2x and so i can clean that my derivative a little bit.

01:43 So i'll write this as f prime is equal to.

01:48 And so the twos are going to cancel.

01:50 So i have x in the numerator.

01:53 And then my denominator is going to be x squared plus six to the positive one half power.

01:59 And we know the one half power is the square root.

02:01 So i'll just write it that.

02:02 Way as the square root of x squared plus 6 in the denominator.

02:09 So i know for part a i'm trying to find my critical numbers.

02:16 So to do that, i set the first derivative equal to zero.

02:19 So if i set f prime equal to zero, that means set the numerator equal to zero.

02:26 So if i set the numerator equal to zero, i get x equals zero.

02:30 So this is a critical number.

02:34 I'm going to abbreviate it as then i also want to check f prime being undefined...

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