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Question

# Which of the following statements is/are true for the function f(x)=x13+7x3âˆ’5x+1

A
f(x) has at least 1 root (real) in [0,1]
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B
f(x) has at least 1 root (real) in R
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C
f(x) has at most 1 root (real) in [0,1]
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D
f(x) has at most 1 real root in R
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Solution

## The correct option is B f′(x) has at least 1 root (real) in Rf(x)=x13+7x3−5x+1 f(0)=1(>0), f(1)=1+7−5+1=4(>0) f(12)=(12)13+78−52+1 (<0) By mean value theorem f(x) has atleast two roots in (0,1). f′(x)=13x12+21x2−5 f′(0)=−5(<0), f′(1)=13+21−5 (>0) f′(−1)=13+21−5 (>0) By M.V.T. f′(x) has atleast two roots in R.

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