Hello, I recently completed this assignment, but I'm not sure if I performed it correctly. Below are two attachments one being the question and the other is me showing my work.

f'(x) = -(x-3)(x+3) = -x2 + 9 Integrate with respect to 'x': f(x) = - 1 2+1 +9x=-2x3 +9x 2 +1 Check to see if - 2x3 +9x satisfies the given properties, f(x) is decreasing at x = -5: Take the derivative of f(x) and evaluate at x = -5. f (x) = -x349 f'(-5) =-(-5)39 =-16 The derivative is negative at x = -5, the function is decreasing at this point. f(x) has a local minimum at x = -3: Take the derivative of f(x) and set to equal zero to find critical points. f'(x) = -x249 -x2 +9 =0 x2 = 9 x =13 Critical points are x = 3 and x = -3. To determine if these are local minima or local maxima, take the second derivative of f(x). f"(x) = -2x f" (x) =-2(-3) =6 The second derivative is positive at x = -3 (local minimum). f(x) has a local maximum at x = 3: Evaluate the second derivative at x = 3 f"(3) =-2(3) =-6 The second derivative is negative at x = 3(local maximum). Conclusion, the function -1x + 9x satisfies the given properties.For this week's discussion. you are asked to generate a continuous and differentiable function f[a:} with the following properties: - f[:c] is decreasing at a: = 5 - f[:c] has a local minimum at :r: = 3 ' f[:c] has a local maximum at: = 3