1. **Problem Statement:** We want to maximize the function $$f(x) = x^2 + 3x$$ where $$x$$ is an integer in the interval $$[0,31]$$ using a Genetic Algorithm (GA). 2. **Basic Structure of Genetic Algorithm:** - **Initialization:** Start with a randomly generated population of candidate solutions (chromosomes), here representing values of $$x$$. - **Selection:** Choose individuals based on their fitness (value of $$f(x)$$) to reproduce. - **Crossover:** Combine pairs of selected individuals to create offspring. - **Mutation:** Introduce small random changes to offspring to maintain genetic diversity. - **Evaluation:** Calculate fitness of new individuals. - **Replacement:** Form a new population from offspring. - **Termination:** Repeat until a stopping criterion is met (e.g., max generations or convergence). 3. **Applying GA to maximize $$f(x) = x^2 + 3x$$:** - **Encoding:** Represent $$x$$ as a 5-bit binary string since $$31_{10} = 11111_2$$. - **Fitness Function:** $$fitness(x) = x^2 + 3x$$. - **Population:** Initialize with random 5-bit strings. - **Selection:** Use roulette wheel or tournament selection based on fitness. - **Crossover and Mutation:** Apply standard GA operators. 4. **Mathematical Analysis:** - The function is a parabola opening upwards. - Vertex formula for $$f(x) = ax^2 + bx + c$$ is $$x = -\frac{b}{2a}$$. - Here, $$a=1$$, $$b=3$$, so vertex at $$x = -\frac{3}{2} = -1.5$$ (outside domain). - Since parabola opens upwards and vertex is negative, function increases over $$[0,31]$$. - Maximum at $$x=31$$ with $$f(31) = 31^2 + 3\times31 = 961 + 93 = 1054$$. 5. **Summary:** - GA will evolve populations towards $$x=31$$. - The maximum value of $$f(x)$$ in the domain is $$1054$$ at $$x=31$$. **Graph Description:** - The graph is a parabola opening upwards. - X-axis: values from 0 to 31. - Y-axis: function values $$f(x)$$. - The curve rises as $$x$$ increases from 0 to 31. Final answer: The maximum value of $$f(x)$$ on $$[0,31]$$ is $$1054$$ at $$x=31$$.