FOr the function f(x)=x^2e^x

Find the critical points, increasing decreasing, first and second derivative and concavity

I am having trouble finding the concavity >_<

First derivative: xe^x(x+2)
Second : e^x(x^2+4x+2)

Critical points from first derivate:
x=0,-2

Increasing: (-infinity,0) (-2, infinity)
Decreasing: (0,-2)

I can't figure out how to do the concavity part of it.

The "concavity" of the function comes from the second derivative, where there is a critical point.

If the second derivative is positive at a point where the first derivative is zero, the function is "concave upward" there. If it is negative, it is concave downward.

Increasing: (-infinity,0) (-2, infinity)

Decreasing: (0,-2)

This does not make sense. Coming from -oo you get to x = -2 before you get to x = 0

Now if I put in a large - number, say -100 for x
I get (-100)^2 e^-100= 10^4 *3.72*10^-44
which sure looks like 0
If I put in x = -2 I get
(-2)^2 e^-2 = .54
so it increases from x = -oo to x = 0

Now at x = 0, we know f(x) is zero, so it is decreasing from x = -2 to x = 0
We can be sure that it then heads up to the right so increasing from x = 0 to x --> +oo

Well, I suppose concave means curving upward. That would happen where the second derivative is +
Now where is the second derivative zero?
x^2+4x + 2 = 0
x = [ -4 +/- sqrt(16 -8) ] /2
= -2 +/- .5 sqrt 8
= -2 +/- sqrt 2
so the second derivative may change sign at -3.41 and at -.586
in other words to the left and right of where the function is level at (-2,.54)

To the left of -3.41 ,say x = -5, the second derivative is +, so curving up, concave from -oo to -3.41
at x = -2, the second derivative is negative, so curves down from -3.41 to - .586
Then at x = 0 the second derivative is + again so it starts curving up at x = -.586 and continues curving up from then on

To find the concavity of the function f(x) = x^2e^x, you need to analyze the second derivative. The concavity can be determined by examining the sign of the second derivative (positive or negative) and the points where it changes sign.

In this case, the second derivative of f(x) is given by:

f''(x) = e^x(x^2 + 4x + 2)

To determine the concavity, follow these steps:

Step 1: Find the critical points of the second derivative.

To find the critical points, set f''(x) equal to zero and solve for x:

e^x(x^2 + 4x + 2) = 0

This equation has no real solutions since e^x is always positive. Therefore, there are no critical points.

Step 2: Choose test points and evaluate f''(x) at these points.

To analyze the concavity, choose test points from each interval identified from the first derivative (increasing and decreasing ranges). Plug these test points into f''(x) and determine if the result is positive or negative.

For example, consider the interval (-∞, 0). Pick a test point within this interval, such as x = -1. Plug this value into f''(x):

f''(-1) = e^(-1)((-1)^2 + 4(-1) + 2) = e^(-1)(1 - 4 + 2) = e^(-1)(-1) < 0

Since f''(-1) is negative, this means the concavity is downward (concave down) in the interval (-∞, 0).

Similarly, test the other intervals determined by the first derivative to find their concavity.

Interval (-2, ∞):
Choose a test point within this interval, such as x = 1. Plug this value into f''(x):

f''(1) = e^1(1^2 + 4(1) + 2) = e(1 + 4 + 2) = 7e > 0

Since f''(1) is positive, this means the concavity is upward (concave up) in the interval (-2, ∞).

To summarize the concavity:

- The function f(x) = x^2e^x is concave down in the interval (-∞, 0).
- The function f(x) = x^2e^x is concave up in the interval (-2, ∞).

Note: It's always a good practice to graph the function or plot the points to visually confirm the concavity.