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If f(x)={{:(x",",0lexle1),(2-e^(x-1)",",...

If `f(x)={{:(x",",0lexle1),(2-e^(x-1)",",1ltxle2),(x-e",",2ltxle3):}` and `g'(x)=f(x), x in [1,3]`, then```

A

g(x) has no local maxima

B

g(x) has no local minima

C

g(x) has local maxima at `x=1+ln2` and local minima at x = e

D

g(x) has local minima at `x=1+ln2` and local maxima at x = e

Text Solution

Verified by Experts

The correct Answer is:
C
`f(x)=g'(x)={{:(2-e^(x-1)",",1ltxle2),(x-2",",2ltxle3):}`
and `x-e=0 rArr x=e`
Graph of f(x) is as shown in the following figure.

From the figure, derivative changes sign from '+' to '-' at `x=1 +ln2` and from `'-'" to "'+'" at "x=e`
Hence `x=1ln2` is point of maxima and `x=e` is point of minima
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