The Derivative a Geometry (a) - Monotone and Extremes

And discriminating monotonicity extremum

Monotonic discrimination

• If y = f (x) have in the interval I f '(x)> 0, then y = f (x) increases strictly monotonically on I
• If y = f (x) has f '(x) in the interval I <0, then y = f (x) increases strictly monotonically on I

Fermat lemma (a necessary condition for extreme points)

• A point-order derivative is a necessary condition for extreme point (the number of extreme values must be turned to 0, the derivative 0 is not necessarily the extreme value, such as X = Y ^{. 3} )
• Provided f (x) = In X X ₀ at derivable, and at point X ₀ to obtain the extremes, it certainly has F '(X ₀ ) = 0

First sufficient condition determining extremum (the first derivative of left and right neighbors)

• Extreme points is not necessarily a guide can point
• Left neighborhood, f '(x) <0 , and the right neighborhood, f' (x)> 0 , then f (x) = x in X ₀ at a local minimum value
• Left neighborhood, f '(x)> 0 , and the right neighborhood, f' (x) <0 , then f (x) = x in X ₀ at a maximum value to obtain
• If f '(x) in the neighborhood of about the same number, the point X ₀ is not extreme point

Second sufficient condition determining extremum (second derivative)

• Provided f (x) at X = X ₀ second-order differentiable, and F '(X ₀ ) = 0, F' '(X ₀ ) ≠ 0
• If F '' (X ₀ ) <0, then f (x) in the X ₀ acquired at the maximum value
• If F '' (X ₀ )> 0, then f (x) in the X ₀ local minimum at
• It can be defined by a first derivative and proof security number

Determining a second extreme value sufficient condition (higher derivative)

• f (x) in the X ₀ may be turned at the n-th order, and F ^(m) (X ₀ ) = 0 (m = 1,2, ...,. 1-n), F ⁽ⁿ⁾ (X) ≠ 0 ( n≥2)
• f'(x₀)=f''(x₀)=...=f^(n-1)(x₀)=0
  
• If n is an even number , and F ⁽ⁿ⁾ (X ₀ ) <0 when, f (x) in the X ₀ acquired at the maximum value
• If n is an even number , and F ⁽ⁿ⁾ (X ₀ )> 0 when, f (x) in the X ₀ acquired at the local minimum value

The Lagrange value theorem (contact function and derivative function)

• f(b) - f(a) = f'(ξ)(b - a)
• f(x) - f(x₀) = f'(ξ)(x - x₀)