$y=\sqrt{x} e^{x}$, then $y^{(2)}=\frac{e^{x}}{\sqrt{x^{3}}}\left(x^{2}+x-\frac{1}{4}\right)$ A. True B. False

Step 1 ：Given the function \(y = \sqrt{x} e^{x}\), we need to find its second derivative \(y^{(2)}\) and check if it matches the given expression \(\frac{e^{x}}{\sqrt{x^{3}}}(x^{2}+x-\frac{1}{4})\).

Step 2 ：First, find the first derivative \(y'\):

Step 3 ：\(y' = \frac{d}{dx}(\sqrt{x} e^{x}) = \frac{1}{2\sqrt{x}} e^{x} + \sqrt{x} e^{x}\)

Step 4 ：Next, find the second derivative \(y^{(2)}\):

Step 5 ：\(y^{(2)} = \frac{d}{dx}(\frac{1}{2\sqrt{x}} e^{x} + \sqrt{x} e^{x}) = \frac{1}{4x\sqrt{x}} e^{x} - \frac{1}{2\sqrt{x}} e^{x} + e^{x}\sqrt{x}\)

Step 6 ：Simplify the second derivative:

Step 7 ：\(y^{(2)} = \frac{e^{x}}{\sqrt{x^{3}}}(x^{2} - \frac{1}{4})\)

Step 8 ：Comparing this with the given expression, we see that they are not equal. Therefore, the statement is \(\boxed{\text{False}}\).