find the value(s) of p for which the function ?
A function #f(x)# is defined about a point #x = 0# by
#f(x) = x + px^2 + O(x^5)# ,
where #p# #in# #RR# . Apply Taylor polynomial about #x = 0# to find the value(s) of #p# for which the function
#g(x) = f(sinx) - x +# #(1)/(3)x^2# + #(1)/(3)x^3# ,
(i) Has a local minimum value ?
(ii) Has a local maximum value ?
(iii) Has an inflection point at #x = 0# ?
A function
where
(i) Has a local minimum value ?
(ii) Has a local maximum value ?
(iii) Has an inflection point at
1 Answer
(i) local minimum if
(ii) local maximum if
(i) inflection point if
Explanation:
Let's first look at the first few terms of the Taylor expansion of the function
Let us now take a look at the Taylor expansion for
Thus if
Since
On the other hand,
will imply that the function will have a local maximum at
If
Note
This could also have been solved by using successive derivatives of