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点击展开 + +**问题分析**: +这是一个标准的二次函数性质分析问题。二次函数的一般形式为 $y = ax^2 + bx + c$。 +本题中,$a = -2, b = 8, c = -3$。 + +**方法选择**: +可以通过配方法将一般式转化为顶点式 $y = a(x-h)^2 + k$,从而直接读出顶点 $(h, k)$ 和最值。 +也可以利用导数法求极值点。 +这里我们使用 SymPy 进行符号计算,通过求导数的方法来确定顶点和最值。 + +**推导过程**: + +1. **求导数**: + 对 $y = -2x^2 + 8x - 3$ 关于 $x$ 求导: + $$ \frac{dy}{dx} = -4x + 8 $$ + +2. **求驻点**: + 令导数为 0,解方程: + $$ -4x + 8 = 0 \implies x = 2 $$ + +3. **求顶点坐标**: + 将 $x = 2$ 代入原函数求 $y$: + $$ y = -2(2)^2 + 8(2) - 3 = -8 + 16 - 3 = 5 $$ + 所以顶点坐标为 $(2, 5)$。 + +4. **判断最值**: + 计算二阶导数: + $$ \frac{d^2y}{dx^2} = -4 $$ + 因为二阶导数小于 0,说明该驻点是极大值点。 + 对于二次函数,极大值即为全局最大值。 + + + +## 5. 📊 运行结果 + +
+点击展开 + +``` +Critical points (x): [2] +Vertex: (2, 5) +Second derivative: -4 +The vertex is a maximum. +Maximum value: 5 +``` + +
diff --git a/20260109_163430_quadratic_vertex工具GithubCopilot模型gemini-3-pro-preview/solve.py b/20260109_163430_quadratic_vertex工具GithubCopilot模型gemini-3-pro-preview/solve.py new file mode 100644 index 0000000..66d6669 --- /dev/null +++ b/20260109_163430_quadratic_vertex工具GithubCopilot模型gemini-3-pro-preview/solve.py @@ -0,0 +1,41 @@ +# /// script +# requires-python = ">=3.11" +# dependencies = ["sympy"] +# /// + +import sympy as sp + +def solve(): + x = sp.symbols('x', real=True) + y = -2*x**2 + 8*x - 3 + + # Find derivative to find critical points + dy_dx = sp.diff(y, x) + critical_points = sp.solve(dy_dx, x) + + print(f"Critical points (x): {critical_points}") + + if not critical_points: + print("No critical points found.") + return + + vertex_x = critical_points[0] + vertex_y = y.subs(x, vertex_x) + + print(f"Vertex: ({vertex_x}, {vertex_y})") + + # Check second derivative to confirm maximum + d2y_dx2 = sp.diff(dy_dx, x) + print(f"Second derivative: {d2y_dx2}") + + if d2y_dx2 < 0: + print("The vertex is a maximum.") + max_val = vertex_y + else: + print("The vertex is a minimum.") + max_val = vertex_y + + print(f"Maximum value: {max_val}") + +if __name__ == "__main__": + solve()