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Question

Find the intervals in which the function f(x)=x44x35x2+24x+12 is
(a) strictly increasing,
(b) strictly decreasing.

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Solution

f(x)=x44x35x2+24x+12

f(x)=x33x210x+24

=(x2)(x4)(x+3)

Put f(x)=0, we have

x=3,2,4

So, the intervals are (,3),(3,2),(2,4) and (4,).

IntervalSign. of f(x)Comment onNature of f(x)(,3)()()() i.e.,<0f(x) is decreasing(3,2)()()(+) i.e.,>0f(x) is increasing(2,4)(+)()(+)i.e.,<0f(x) is decreasing(4,)(+)(+)(+)i.e.,>0f(x) is increasing


Hence, f(x) is strictly increasing in (3,2)(4,),
and f(x) is strictly decreasing in (,3)(2,4).

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