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Consider the function f(x)=x3px2+qx defined on [1,3]. If f satisfies the hypothesis of Rolle's theorem such that c=32, then the value of 4p+q+16 is 


Solution

f(x)=x3px2+qx
f satisfies the hypothesis of Rolle's theorem.
f(1)=f(3)
1p+q=279p+3q
8p2q=26
4pq=13     (1)

Also, f(c)=0
3c22pc+q=0
3(32)22p(32)+q=0
2712p+4q=0     (2)

Solving (1) and (2), we get
p=254, q=12

4p+q+16=25+12+16=53

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