Calculus Volume 3 · 免费预览
最大值与最小值问题
二元函数的局部极值出现在梯度为零或不存在的临界点处，可借助二阶导数判别法的判别式进行分类；而在闭的有界区域上求绝对极值时，则需考察其内部和边界。
1. Critical points
Critical points
2. The second derivative test
The second derivative test
3. Absolute extrema on closed regions
Absolute extrema on closed regions
4. Applications
Applications
5. Critical point condition
For a differentiable function of two variables, local extrema occur only where which condition holds?
6. Saddle point
A saddle point of f(x, y) is a critical point which is what?
7. Discriminant formula
The second derivative test's discriminant D equals which expression?
8. Positive discriminant minimum condition
A critical point with D positive is a local minimum when which further condition holds?
9. D equals zero case
When the second derivative test's discriminant D equals zero at a critical point, what does the test conclude?
10. Critical point count paraboloid
The paraboloid f = x^2 + y^2 has one critical point. As a number, how many? Answer with a number.
11. Saddle discriminant sign
For the saddle x^2 - y^2, the discriminant D is negative. As a sign, what is it? Answer with the single character.
12. Hessian name
The matrix of second partials is named for which mathematician, surname only? Answer in lowercase.
13. Least squares connection
The regression line's least squares fit minimizes the sum of squared errors, and its slope and intercept are found by which condition?
14. Boundary candidate source
On a closed bounded region, the absolute extrema's candidates include interior critical points and the boundary's extrema, found by which methods? Answer with two words.