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Question

Find the points at which the function f given by f(x)=(x2)4(x+1)3 has
(i) local maxima
(ii) local minima
(iii) point of reflexion.

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Solution

Consider the given expression,

f(x)=(x2)4(x+1)3 …….(1)

Differentiate with respect to x we get

f(x)=(x2)43(x+1)2+(x+3)34(x2)3 …(2)

For maxima and minima

f(x)=0

(x2)43(x+1)2+(x+3)34(x2)3=0

(x2)3(x+1)2[3(x2)+(x+3)4]=0


(x2)3(x+1)2[3x6+4x+12]=0

(x2)3(x+1)2(7x+6)=0

x=2,x=1,x=67

At point x=2,x=1 f(x)=0 hence these are point of inflection.

And x=67 ,f′′(x) is positive ,then function will be maximum.so this is the point of local maxima.


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