The number of salmon swimming upstream to spawn is approximated by the following function, where \( x \) represents the temperature of the water in degrees Celsius. Find the water temperature that produces a local maximum number of salmon swimming upstream. \[ S(x)=-x^{3}+10.5 x^{2}+234 x+5197,4 \leq x \leq 20 \]

To find the water temperature that produces a local maximum number of salmon swimming upstream, we need to find the critical points of the function \( S(x) = -x^{3} + 10.5x^{2} + 234x + 5197 \) within the given interval \( 4 \leq x \leq 20 \). Critical points are found by taking the derivative of the function and setting it equal to zero. The derivative of \( S(x) \) with respect to \( x \) is: \[ S'(x) = -3x^{2} + 21x + 234 \] Now, we need to solve the equation \( S'(x) = 0 \) to find the critical points. Let's solve the equation \( -3x^{2} + 21x + 234 = 0 \) to find the critical points. Solve the quadratic equation by following steps: - step0: Solve by factoring: \(-3x^{2}+21x+234=0\) - step1: Factor the expression: \(-3\left(x-13\right)\left(x+6\right)=0\) - step2: Divide the terms: \(\left(x-13\right)\left(x+6\right)=0\) - step3: Separate into possible cases: \(\begin{align}&x-13=0\\&x+6=0\end{align}\) - step4: Solve the equation: \(\begin{align}&x=13\\&x=-6\end{align}\) - step5: Rewrite: \(x_{1}=-6,x_{2}=13\) The critical points are \( x = -6 \) and \( x = 13 \). Since the given interval is \( 4 \leq x \leq 20 \), we only consider the critical point \( x = 13 \) within this interval. To determine if this critical point corresponds to a local maximum, we need to examine the behavior of the function around this point. We can do this by looking at the second derivative of the function, \( S''(x) \), and determining its sign at the critical point. The second derivative of \( S(x) \) is: \[ S''(x) = -6x + 21 \] Now, we need to evaluate \( S''(x) \) at \( x = 13 \) to determine the nature of the critical point. Calculate the value by following steps: - step0: Calculate: \(-6\times 13+21\) - step1: Multiply the numbers: \(-78+21\) - step2: Add the numbers: \(-57\) The second derivative at \( x = 13 \) is \( -57 \), which is negative. This indicates that the critical point \( x = 13 \) corresponds to a local maximum. Therefore, the water temperature that produces a local maximum number of salmon swimming upstream is \( x = 13 \) degrees Celsius.