Calculus Volume 1 · 免费预览
极大值与极小值
函数在一个区间上的极值可在临界点和端点中求得，并依据导数判别法加以分类；极值定理则保证极值在闭且有界的区间上存在。
1. Extrema and the extreme value theorem
Extrema and the extreme value theorem
2. Critical points and the closed interval method
Critical points and the closed interval method
3. The first derivative test
The first derivative test
4. The second derivative test
The second derivative test
5. Fermat's theorem conclusion
Fermat's theorem on local extrema applies to an interior point c where f is differentiable and has a local extremum. What conclusion does it draw?
6. Critical point definition
Which points belong on the critical point list of a function f?
7. Closed interval method
To find the absolute extrema of a continuous function on [a, b], which values must be compared?
8. First derivative test sign change
At a critical point c, the derivative f prime changes from positive to negative as x passes through c. What does the first derivative test conclude?
9. Second derivative test failure
The second derivative test examines f double prime of c at a stationary point c. What does f double prime of c = 0 tell us?
10. No guarantee on open intervals
The function f(x) = x on the open interval (0, 1) has no maximum or minimum there. Which hypothesis of the extreme value theorem fails?
11. Cubic critical points
The function f(x) = x^3 - 3x has derivative 3x^2 - 3. Besides x = 1, what is its other critical point? Answer with a number.
12. Absolute minimum value
On the interval [0, 2], the values of f(x) = x^3 - 3x at the candidates are f(0) = 0, f(1) = -2, and f(2) = 2. What is the absolute minimum value? Answer with a number.
13. Stationary point count
The function f(x) = x^4 - 2x^2 has derivative 4x(x^2 - 1). How many stationary points does it have? Answer with a number.
14. Inconclusive test count
For f(x) = x^4, both f prime and f double prime vanish at c = 0, so the second derivative test is inconclusive, yet the point is a local minimum. How many local extrema does x^4 have on the whole real line? Answer with a number.