Using the second derivative, it is found that the graph is concave down on the interval \((-\infty, -2)\).
A function is concave down when the second derivative is negative.
The function is:
\(f(x) = 2xe^x\)
The first derivative is as follows, applying the product rule:
\(f^{\prime}(x) = 2e^x + 2xe^x\)
The second derivative is the derivative of the first derivative, given by:
\(f^{\prime\prime}(x) = 2e^x + 2e^x + 2xe^{x} = 4e^x + 2xe^x = e^x(4 + 2x)\)
The exponential is always positive, so the second derivative is negative if:
\(4 + 2x < 0\)
\(2x < -4\)
\(x < -\frac{4}{2}\)
\(x < -2\)
Hence, the graph is concave down on the interval \((-\infty, -2)\).
A similar problem is given at https://brainly.com/question/13539822
