Let f(x)=2 e^x-6 x. a. Find all points on the graph of f at which the tangent line is horizontal

Let f(x)=2 e^x-6 x. a. Find all points on the graph of f at...

Key Concepts

Derivative

The derivative represents the instantaneous rate of change of a function with respect to its variable. It is foundational in calculus and is used to determine the slope of the tangent line to the graph at any given point. Calculating the derivative enables the study of how functions behave and change locally.

Tangent Lines and Their Slopes

A tangent line to a function’s graph touches the curve at a single point and has the same slope as the function at that point. Understanding tangent lines, especially when they are horizontal (slope zero) or have a specified slope, is crucial for analyzing and approximating the local behavior of functions.

Differentiation of Exponential Functions

Exponential functions, such as those involving e^x, have unique rules for differentiation. The process includes applying the standard derivative rules for exponential functions, which is essential when the function being analyzed contains an exponential term. This concept is critical for correctly computing derivatives when exponential expressions are involved.

Solving Equations Involving Derivatives

After computing the derivative, equations are often set up by equating the derivative to a specific value (such as zero or another constant) to find points where the function meets certain slope conditions. This method is important in the analysis of functions to determine critical points, points of inflection, and specific behaviors of the graph.

Transcript

00:01 Hello, so first we take our given function and compute its derivative, and we find that f prime of x is going to be equal to while we get two.

00:12 So our function is two, e to the x minus six prime.

00:16 So that's going to be two times e to the x and then minus six.

00:22 Okay, so there is our derivative f prime.

00:25 And we take our derivative, we set it equal to zero.

00:28 So we add six to both sides, and we have two.

00:31 Times e to the x is equal to 6, dividing 3 by 2 gives us that e to the x is equal to 3.

00:38 Okay, so if e to the x is equal to 3, that x is then going to be equal to the natural log of 3.

00:48 So we then substitute in the natural log of 3 for x in our function, and we have f of the natural log of 3 is going to be equal to 2 times 3.

01:01 Minus six times the natural log of 3.

01:03 So that's equal to 6 minus 6 times the natural log of 3.

01:09 So therefore, the point at which the tangent line is horizontal is going to be at the point natural log of 3, comma, and then the other is the first, there's the x coordinate, and the y coordinate is going to be 6 minus 6 times the natural log of 3.

01:29 There's a point.

01:30 Ln of 3, comma 6 minus 6 times the network of 3.

01:35 Okay, then for part b, we want to set our function, or set our derivative equal to 12.

01:43 So we have f prime equal to 12.

01:45 So we have 2 times e to the x minus 6 equal to 12...

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