Step 1 :Given the function \(f(x)=x^{2} e^{c x}\), we are asked to find the value of \(c\) for which \(f\) has a critical point at \(x=1\).
Step 2 :A critical point of a function occurs where its derivative is zero or undefined. So, we first need to find the derivative of the function.
Step 3 :The derivative of the function \(f(x)=x^{2} e^{c x}\) is \(f'(x) = c*x^{2}*e^{c x} + 2*x*e^{c x}\).
Step 4 :We set the derivative equal to zero and solve for \(x\). The solutions are \(x = 0\) and \(x = -2/c\).
Step 5 :We substitute \(x=1\) into the equation and solve for \(c\). The solution is \(c = -2\).
Step 6 :Final Answer: The value of \(c\) for which \(f\) has a critical point at \(x=1\) is \(\boxed{-2}\).