Determining Points of Inflection for a Function
Key Questions
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#f(x)=x^3+x# By taking derivatives,
#f'(x)=3x^2+1# #f''(x)=6x=0 Rightarrow x=0# ,which is the
#x# -coordinate of a possible inflection point. (We still need to verify that#f# changes its concavity there.)Use
#x=0# to split#(-infty,\infty)# into#(-infty,0)# and#(0,infty)# .Let us check the signs of
#f''# at sample points#x=-1# and#x=1# for the intervals, respectively.
(You may use any number on those intervals as sample points.)#f''(-1)=-6<0 Rightarrow f# is concave downward on#(-infty,0)# #f''(1)=6>0 Rightarrow f# is concave upward on#(0,infty)# Since the above indicates that
#f# changes its concavity at#x=0# ,#(0,f(0))=(0,0)# is an inflection point of#f# .I hope that this was helpful.
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No. Consider
#f(x)=x# - this function's concavity does not change throughout the entire run of the function.All polynomials with odd degree of 3 or higher have points of inflection, and some polynomials of even degree (again, higher than 3) have them. The best way to determine if a function has a point of inflection is to look at its second derivative - if the second derivative can equal zero, the original function has a point of inflection.
Questions
Graphing with the Second Derivative
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Relationship between First and Second Derivatives of a Function
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Analyzing Concavity of a Function
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Notation for the Second Derivative
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Determining Points of Inflection for a Function
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First Derivative Test vs Second Derivative Test for Local Extrema
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The special case of x⁴
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Critical Points of Inflection
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Application of the Second Derivative (Acceleration)
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Examples of Curve Sketching