9. Find the local maximums, local minimums, domain, and range for: \[ f(x)=0.5 x^{3}-3 x^{2}+1.5 \]

We are given the function   f(x) = 0.5x³ – 3x² + 1.5 Step 1. Find the first derivative f '(x):   f '(x) = d/dx [0.5x³ – 3x² + 1.5]        = 1.5x² – 6x Factor the derivative:   1.5x² – 6x = 1.5x(x – 4) Step 2. Set the derivative equal to zero to find critical points:   1.5x(x – 4) = 0 Thus, x = 0 or x = 4. Step 3. Determine whether these critical points are local minima or maxima using the second derivative test. Find the second derivative:   f ''(x) = d/dx [1.5x² – 6x] = 3x – 6 Evaluate f ''(x) at the critical points: — At x = 0:   f ''(0) = 3(0) – 6 = –6, which is less than 0. So f has a local maximum at x = 0.   Calculate f(0) = 0.5(0)³ – 3(0)² + 1.5 = 1.5.   Thus, the local maximum is at (0, 1.5). — At x = 4:   f ''(4) = 3(4) – 6 = 12 – 6 = 6, which is greater than 0. So f has a local minimum at x = 4.   Calculate f(4) = 0.5(4)³ – 3(4)² + 1.5           = 0.5(64) – 3(16) + 1.5           = 32 – 48 + 1.5           = –14.5.   Thus, the local minimum is at (4, –14.5). Step 4. Domain and Range Domain: Since f(x) is a polynomial, the domain is all real numbers: (–∞, ∞). Range: f(x) is a cubic function. Cubic functions with a nonzero x³ term always have a range of (–∞, ∞). Summary: • Local Maximum: (0, 1.5) • Local Minimum: (4, –14.5) • Domain: (–∞, ∞) • Range: (–∞, ∞)