The function f(x) = e^3x +6x^2 +1 has a horizontal tangent at x =?

In summary, the conversation discusses solving an equation involving the second derivative and using a calculator to calculate the horizontal tangent and further derivatives to determine the nature of the solution.
  • #1
meredith
16
0

Homework Statement




i can't solve it!
im really lost. i know you find f''(x) (i got 9e^3x +12) but i don't know where to go from there
what would i do on my calculator?
 
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  • #2
to calculate the horizontal tangent you only need to solve for f ' (x) = 0. f '' (x) is used and further derivatives are used to check whether that particular solution of the first equation gives a maxima, minima or an inflexion point.

Solve 3e^(3x) + 12x = 0
 
  • #3
And that equation you will need to solve numerically, perhaps by graphing it on your graphing calculator and "zooming" in where it crosses the x-axis.
 

1. What is the x-coordinate of the point where the function f(x) has a horizontal tangent?

The x-coordinate of the point where the function f(x) has a horizontal tangent is approximately -0.328.

2. How do you determine if a function has a horizontal tangent at a certain point?

To determine if a function has a horizontal tangent at a certain point, you can take the first derivative of the function and set it equal to 0. If the resulting equation has a solution for the x-coordinate of the point, then the function has a horizontal tangent at that point.

3. What does it mean for a function to have a horizontal tangent?

When a function has a horizontal tangent at a certain point, it means that the slope of the tangent line at that point is equal to 0. This indicates that the function is neither increasing nor decreasing at that point, and the graph of the function has a flat spot at that point.

4. Can a function have more than one horizontal tangent?

Yes, a function can have more than one horizontal tangent. This occurs when the first derivative of the function has multiple solutions for the x-coordinate of the points where the function has a horizontal tangent.

5. How can you use the graph of a function to determine the x-coordinate of the point where it has a horizontal tangent?

To determine the x-coordinate of the point where a function has a horizontal tangent, you can look for the point on the graph where the slope of the tangent line is equal to 0. This will be the point where the function has a horizontal tangent.

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