The Derivative learn geometry applications (2) - Convexity of the inflection point

Concavity and convexity

 

Inflection point

• Convex arc and concave arc cut-off point
• Inflection point on the curve, writing (X ₀ , F (X ₀ ) )
• The extreme points on the domain, writing X ₀

Convexity of discrimination

Second Order derivable inflection point is a necessary condition

First sufficient condition determination of irregularities (neighborhoods around the second derivative of opposite sign)

• X ₀ in a neighborhood to the heart, the second derivative is present, at which point the second derivative of both sides of the variable number (from positive to negative, or from the positive to negative), the point (X ₀ , F (X ₀ )) of the knee of the curve

Determining a second sufficient condition of irregularities (second derivative = 0, the third derivative ≠ 0)

• X ₀ in a neighborhood to the heart of the third-order differentiable, and f '' (x0) = 0 , f '' '(x0) ≠ 0, then (X0, F (X ₀ )) as an inflection point
• And using the derivative defined proof security number

The third judgment sufficient condition of irregularities

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