On what interval(s) is the curve y=(2e^x)-(e^(-3x)) concave downward?
To determine the interval(s) where the curve y = 2e^x - e^(-3x) is concave downward, we need to find the second derivative and analyze its sign.
First, let's find the first derivative of the function:
y = 2e^x - e^(-3x)
To differentiate, we use the chain rule:
dy/dx = d/dx(2e^x) - d/dx(e^(-3x))
= 2(d/dx(e^x)) - (-3)(d/dx(e^(-3x)))
= 2e^x + 3e^(-3x)
Now, let's find the second derivative by differentiating the first derivative:
d^2y/dx^2 = d/dx(2e^x + 3e^(-3x))
= 2(d/dx(e^x)) + 3(d/dx(e^(-3x)))
= 2e^x - 9e^(-3x)
Now that we have the second derivative, we can analyze its sign to determine when the function is concave downward. For the curve to be concave downward, the second derivative must be negative.
Setting the second derivative to less than zero:
2e^x - 9e^(-3x) < 0
To solve this inequality, we can factor it as follows:
e^(-3x)(2e^4x - 9) < 0
Now we have two factors: e^(-3x) and (2e^4x - 9). For the inequality to be true, either both factors need to be negative or one of them should be zero.
1. e^(-3x) < 0:
Exponential functions are always positive, so this factor is never negative.
2. 2e^4x - 9 < 0:
Let's solve this inequality separately:
2e^4x < 9
e^4x < 4.5
Since e^4x is always positive, this inequality does not have any real solutions.
Therefore, there are no intervals where the curve y = 2e^x - e^(-3x) is concave downward.
Set the second derivative equal to zero and solve for x.
Test the regions below and above x to determine which interval(s) yield a negative output.