Application of Derivatives Part 2

Let f(x)=2x5+5x4+10x3+10x2+10x+10
Using hit and trial method,
f(−2)=−34<0 and f(−1)=3>0
Hence, f(x) has a root in (−2,−1).
Again,
f′(x)‌‌=10x4+20x3+30x2+20x+10
‌‌=10x2(x2+2x+3+‌

2

x

+‌

1

x2

)
‌‌=10x2[(x2+‌

1

x2

)+2(x+‌

1

x

)+3]
‌‌=10x2[(x+‌

1

x

)2+1+2(x+‌

1

x

)]
‌‌=10x2[(x+‌

1

x

+1)2]>0‌∀x
⇒f(x) is strictly increasing function, since degree of f(x) is odd.
∴ It has exactly one real root.
Therefore, f(x) has atleast one root in
⇒‌‌|a|=|−2|=2