Find the critical numbers of the function. g(x)=4 x^2(3^x) |...

Name: Problem 18, Chapter 4: Calculus Early Transcendental Functions
Uploaded: 2021-08-17T16:38:53-08:00
Duration: 4 min 3 s
Channel: Diane Koenig
Description: Find the critical numbers of the function. g(x)=4 x^2(3^x)

Key Concepts

Exponential Functions

Exponential functions involve a constant base raised to a variable exponent, and they have unique differentiation properties. Their derivatives typically involve the original exponential function multiplied by the logarithm of the base.

Derivative

The derivative represents the instantaneous rate of change of a function. It is a fundamental tool in calculus used to determine where functions increase or decrease, and to find slope values at particular points.

Critical Numbers

Critical numbers of a function are values of the input where the derivative is either zero or undefined. They are often found when solving for the zeros of the derivative and are potential locations for local maxima, local minima, or inflection points.

Product Rule

The product rule is a method used for differentiating functions that are products of two or more functions. It states that the derivative of a product is the derivative of the first function times the second function plus the first function times the derivative of the second function.

Transcript

00:02 Here we are asked to find the critical numbers for the function g of x equals 4x squared times 3d of x.

00:08 Now first we want to look at what the domain of this original function is, because critical numbers are values in the domain of the original function for which the first derivative is zero or undefined.

00:20 So we always want to be aware of the domain of the original function.

00:24 Will our work with our 4x squared times 3 to the x, any real number can be squared times 4.

00:32 The base of three to any real number.

00:36 So this domain is all reals.

00:42 So any values that i get for the derivative that's zero or undefined is in the domain of the original function.

00:49 And so i'm good with that.

00:51 So now next up, we're going to take our derivative.

00:54 So g prime of x.

00:57 This is a product.

00:58 So we have to use a product rule.

01:00 We have a number times a letter base to a number power.

01:03 That'll use the power rule, but then i have a number base to a letter power.

01:07 So that's a exponential function, and i need to use the derivative for exponential functions for that.

01:13 Now remember, differentiating a product, it's derivative of the first.

01:17 So derivative of 4x squared is 8x times the second factor, 3 to the x, plus the first 4x squared times the derivative of the second factor.

01:28 And the derivative of 3 to the x, the derivative of an exponential is the same exponential times the derivative of the exponent, the derivative of x is 1, and then if the base is not a, you also have to multiply by the natural log of the base...

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