Find the local maxima of f(x) = -x^2 + 6x - 8. (2022)

Find the local maxima of f(x) = -x^2 + 6x - 8. (2022)

Step 1: Write down the function f(x) = -x^2 + 6x - 8.
Step 2: Find the derivative of the function, which is f'(x). The derivative tells us the slope of the function.
Step 3: Calculate the derivative: f'(x) = -2x + 6.
Step 4: Set the derivative equal to zero to find critical points: -2x + 6 = 0.
Step 5: Solve for x: Add 2x to both sides to get 6 = 2x, then divide both sides by 2 to find x = 3.
Step 6: To find the local maxima, substitute x = 3 back into the original function f(x).
Step 7: Calculate f(3): f(3) = -3^2 + 6(3) - 8.
Step 8: Simplify the calculation: f(3) = -9 + 18 - 8 = 1.
Step 9: The local maximum occurs at x = 3, and the maximum value is f(3) = 1.

Finding Local Maxima – This involves taking the derivative of a function, setting it to zero to find critical points, and evaluating the function at those points to determine local maxima or minima.
Quadratic Functions – Understanding the properties of quadratic functions, including their parabolic shape and the significance of the coefficients in determining the direction of the parabola.