Suppose that h is a function with h'(x) = xe^x. Find all intervals where the function h is concave up.
Now, I understand that concavity is determined by the second derivative, so what my question here is, is it just taking the first derivative and then taking it again? Is it that simple?
Also, does it just involve the Product rule when taking the first derivative?
Would it be:
y' = 1e^x + xe^x *1
How would you approach a second derivative?
y'' = e^x +1e^x +xe^x*1
y'' = 2e^x+xe^x
How would you determine concavity if you can't find where y'' = 0?
you already have h', the first
just take one more
x e^x + e^x = e^x (x+1)
e^x always +
so where x + 1 is +
x> -1
To find the intervals where the function h(x) is concave up, you are correct that you need to examine the second derivative of h. Let's go through the steps:
1. First, find the first derivative of h(x) using the product rule. You correctly applied the product rule in your calculation:
h'(x) = 1e^x + xe^x * 1
= e^x + xe^x
2. Then, to find the second derivative, differentiate the first derivative with respect to x:
h''(x) = (e^x)' + (xe^x)'
= e^x + e^x + xe^x * 1
= 2e^x + xe^x
Now, to determine where the function h(x) is concave up, you need to find the intervals where h''(x) is positive (greater than 0).
Since the exponential function e^x is always positive for any value of x, we only need to consider the sign of 2e^x + xe^x. To do this, we cannot directly set the second derivative equal to zero because it won't indicate the concavity of the function.
Instead, we look for when the second derivative is greater than zero (positive). If 2e^x + xe^x > 0, then the function h(x) is concave up in that interval.
To solve this inequality, we can factor out the common factor e^x:
e^x(2 + x) > 0
Now, we can set each factor equal to zero and find the critical points:
e^x = 0 (No real solutions)
2 + x = 0
x = -2
This means that the second derivative is positive and h(x) is concave up for x < -2 (to the left of -2) and for x > -2 (to the right of -2). So, the function h(x) is concave up in the intervals (-∞, -2) and (-2, ∞).
In summary, to determine the concavity of a function, you need to find the second derivative and analyze its sign. If you cannot find where the second derivative equals zero, you can examine the sign of the expression to determine the concavity.