Question 11
Let $\mathrm{f}$ be the function given below, where $\mathrm{c}$ is a constant. For what value of $c$ does $f$ have a critical point at $x=1$ ?
\[
f(x)=x^{2} e^{c x}
\]
a. -1
b. 0
c. 1
d. -2

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 \(\boxed{-2}\).