Questions: Let f(x) = 400x - 11x^2 - 9. Find the maximum value of f to four decimal places graphically. The maximum value of f is □ . (Round to four decimal places as needed.)

Let f(x) = 400x - 11x^2 - 9. Find the maximum value of f to four decimal places graphically. The maximum value of f is □ . (Round to four decimal places as needed.)
Transcript text: Let $f(x)=400 x-11 x^{2}-9$. Find the maximum value of $f$ to four decimal places graphically. The maximum value of $f$ is $\square$ . (Round to four decimal places as needed.)
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Solution

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Solution Steps

To find the maximum value of the function \( f(x) = 400x - 11x^2 - 9 \) graphically, we can follow these steps:

  1. Plot the function \( f(x) \) over a reasonable range of \( x \) values.
  2. Identify the vertex of the parabola, as it represents the maximum point for a downward-opening parabola.
  3. Use Python to find and display the maximum value to four decimal places.
Step 1: Identify the Function and Its Derivative

We are given the function: \[ f(x) = 400x - 11x^2 - 9 \]

To find the maximum value of \( f(x) \), we first need to find its critical points by taking the derivative and setting it to zero.

Step 2: Compute the Derivative

The derivative of \( f(x) \) is: \[ f'(x) = \frac{d}{dx}(400x - 11x^2 - 9) \] \[ f'(x) = 400 - 22x \]

Step 3: Find the Critical Points

Set the derivative equal to zero to find the critical points: \[ 400 - 22x = 0 \] \[ 22x = 400 \] \[ x = \frac{400}{22} \] \[ x \approx 18.1818 \]

Step 4: Determine the Nature of the Critical Point

To determine whether this critical point is a maximum, we can use the second derivative test. Compute the second derivative of \( f(x) \): \[ f''(x) = \frac{d}{dx}(400 - 22x) \] \[ f''(x) = -22 \]

Since \( f''(x) = -22 \) is negative, the function \( f(x) \) has a local maximum at \( x \approx 18.1818 \).

Step 5: Calculate the Maximum Value

Substitute \( x \approx 18.1818 \) back into the original function to find the maximum value: \[ f(18.1818) = 400(18.1818) - 11(18.1818)^2 - 9 \] \[ f(18.1818) \approx 400(18.1818) - 11(330.5785) - 9 \] \[ f(18.1818) \approx 7272.7272 - 3636.3635 - 9 \] \[ f(18.1818) \approx 3627.3637 \]

Final Answer

The maximum value of \( f \) is: \[ \boxed{3627.3637} \]

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