Math Problem Statement
Solution
From the image, you're solving a calculus problem involving a function , and you are working through finding the derivative , determining where the tangent is horizontal, and calculating the corresponding points.
Let's break it down:
-
Find the derivative : Given the function: The derivative is: You correctly found the derivative, as shown in the image.
-
Find where the tangent line is horizontal: A horizontal tangent line occurs when . So, you need to solve: Dividing the entire equation by 3 simplifies this to: Factor the quadratic: So, or .
-
Find the points on the curve: Now, plug these values of back into the original function to find the corresponding -coordinates.
-
For : So, one point is .
-
For : So, the other point is .
-
Summary:
- The correct values of where the tangent is horizontal are and .
- The corresponding points on the curve are and , which are correct as indicated in the image.
Would you like further details on solving these types of problems or another question breakdown?
Here are five related questions:
- How do you determine if a function has a horizontal tangent line?
- What are the steps to factor a quadratic equation like ?
- What are the implications of the second derivative at the points of horizontal tangency?
- How do you verify if the points found are maxima, minima, or points of inflection?
- What changes in the process if the function includes trigonometric terms?
Tip: Always check if a quadratic equation can be factored before using the quadratic formula—it can save you time!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Quadratic Equations
Formulas
y'(x) = 3x^2 + 6x - 9
Horizontal tangency occurs when y'(x) = 0
Quadratic equation: x^2 + 2x - 3 = 0
Factor: (x - 1)(x + 3) = 0
Theorems
Derivative test for horizontal tangents
Quadratic factoring
Suitable Grade Level
Grade 12 and introductory college-level Calculus
Related Recommendation
Find Horizontal Tangents for y = x^3 + 3x^2 − 9x + 6
Find Points with Horizontal Tangents on Curve y = x^3 + 3x^2 − 9x + 4
Find the Point Where the Tangent Line Is Horizontal for f(x) = 1/3x^3 - 3x^2 + 9x - 9
Finding Horizontal Tangents for the Function f(x) = 2x^2 - 3x + 3
Find Points with Horizontal Tangent Lines and Slope 54 for f(x) = 2x^3 - 6x^2 - 90x - 3