# Is Curve x=3t^2, y=e^2t+1 Concave Up/Down at (3,e^3)?

• rcmango
In summary, to determine the concavity of the curve x = 3t^2, y = e^2t+1 at the point (3, e^3), you will need to take the double derivative. Plug in the values of x and y at the given point and use them to find the concavity of the curve. If you are having trouble with the t variable, try to eliminate it and write y as a function of x.
rcmango

## Homework Statement

Determine whether the curve x = 3t^2, y = e^2t+1 is concave up or concave down at the point (3, e^3)

thanks for any help with this one.

## The Attempt at a Solution

know i need to take the double derivative. Also that would be 6
and 4e^2t+1

what point do i put back into the double derivative. Also, do I use 3 and e^3? they represent x, y, so do they plug into the equation appropriately, or back into t? please help.

If you know how to determine concavity or concavity when given a function y(x), then I suggest you try to "eliminate" that nasty t and try to write y as a function of x!

## What is the equation for the curve in question?

The equation is x=3t^2, y=e^2t+1.

## What does it mean for a curve to be concave up/down?

Concave up means that the curve is shaped like a cup, with the bottom being the lowest point. Concave down means that the curve is shaped like a dome, with the highest point being at the bottom.

## How can I determine if the curve is concave up/down at a specific point?

To determine if the curve is concave up/down at a specific point, you can take the second derivative of the equation and plug in the coordinates of the point. If the result is positive, the curve is concave up. If the result is negative, the curve is concave down.

## Is the curve in question concave up or concave down at (3,e^3)?

To determine this, we need to take the second derivative of the equation. The second derivative is 6, which is positive. Therefore, the curve is concave up at (3,e^3).

## Can the curve be both concave up and concave down at the same time?

No, the curve can only be one or the other at a specific point. However, it is possible for a curve to switch from being concave up to concave down or vice versa at different points along the curve.

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