- #1
nikozm
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Hi,
If f'(x) >= f'(y) can we say that f''(x) >= f''(y) also holds ? And if yes under which conditions ?
Thanks
If f'(x) >= f'(y) can we say that f''(x) >= f''(y) also holds ? And if yes under which conditions ?
Thanks
is not correct. For anything below those bounds it is right. The inequality you stated is only true under certain bounds because the two derivatives vary and may periodically become greater or smaller as in the sine and cosine functions. Bounds the inequality is true under need to be specified , to answer questions about the examples of the inequality you stated. Also because the derivative is a function it varies inequalities can't though and become false for certain values of the function. (The sine cosine remark serves as a good example, but I'm leaning towards arbitrary functions varying in all different manners not just varying periodically.)nikozm said:Hi,
If f'(x) >= f'(y) can we say that f''(x) >= f''(y) also holds ? And if yes under which conditions ?
Thanks
The relation between inequalities for first and second derivatives is known as the second derivative test. It is used to determine the concavity and convexity of a function and whether a critical point is a maximum or minimum point.
The second derivative test involves taking the second derivative of a function and evaluating it at a critical point. If the second derivative is positive, the critical point is a minimum point. If the second derivative is negative, the critical point is a maximum point.
No, the first derivative test can only determine whether a critical point is a local maximum or minimum point. It cannot determine the nature of the critical point (maximum or minimum) like the second derivative test can.
If the first derivative is positive, the function is increasing and the concavity is upward (convex). If the first derivative is negative, the function is decreasing and the concavity is downward (concave).
Yes, the second derivative test can also be used to identify inflection points. If the second derivative changes sign at a critical point, then that critical point is an inflection point.