List the critical numbers of the following function in increasing order. Enter \( N \) in any blank that you don't need to use. \[ f(x)=7 x e^{7 x} \]

**Step 1: Find the derivative.** Given the function \[ f(x) = 7x e^{7x}, \] we differentiate using the product rule. Let \[ g(x)=7x \quad \text{and} \quad h(x)=e^{7x}. \] Then \[ f'(x)=g'(x)h(x)+g(x)h'(x). \] We compute: \[ g'(x)=7, \] and \[ h'(x)=7e^{7x}. \] Thus, the derivative is: \[ f'(x)=7 \cdot e^{7x} + 7x \cdot 7e^{7x} = 7e^{7x} + 49xe^{7x}. \] **Step 2: Factor the derivative.** Factor out the common factor \(7e^{7x}\): \[ f'(x)=7e^{7x}(1+7x). \] **Step 3: Set the derivative equal to zero and solve.** We need to solve: \[ 7e^{7x}(1+7x)=0. \] Since \(7\) and \(e^{7x}\) are never zero for any real \(x\), we set: \[ 1+7x=0. \] Solving for \(x\): \[ 7x=-1 \quad \Rightarrow \quad x=-\frac{1}{7}. \] **Step 4: State the critical number.** The critical number of the function is \[ \boxed{-\frac{1}{7}}. \] Since this is the only critical number, any other blank should be filled with \( N \).