Hi, I just wanted to make sure that I explained this correctly. Can you help?

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[;c} with the following properties: - f[:13] is decreasing at a: = 5 - f{a:) has a local minimum at a: = 3 - f[:13] has a local maximum ata: = 3 Your classmates may have different criteria for their functions, so in your initial post in Brightspace be sure to list the criteria for your function. Hints: - Use calculus! - Before specifying a function f(x) first determine requirements for its derivative f" (3:). For example, one ofthe requirements is that y"f {3) = D. - If you want to find a function g[:r:) such that g[Q) = U and 9(8) = I], then you could try eta?) = (33+ 9) {93' 3)- - If you have a possible function for f [m] then use the techniques in Indefinite Integrals this Module to try a possible f(m). You can generate a plot of your function by clicking the plotting option (the page option with a "P" next to your function input). You may want to do this before clicking \"How Did I 00?". Notice that the label "3(at) =\" is already provided for you. lOnce you are ready to check your function, click "How Did I Do?" below {unlimited attempts). Please note that the bounds on the m-axis go from 6 to ES. It is recommended that you put a multiplication symbol between variables or between a variable and W (should you use it}. Example: Write sin (1r - a3)instead of sin (7m). f (:3) = 1.-'3*)c\"3+9*x @