Application of DerivativeMedium
Question
If the equation anxn + an-1xn-1 + .... + a1x = 0 has a positive root x = α, then the equation nanxn-1 + (n - 1) an-1xn-2 + ..... + a1 = 0 has a positive root, which is-
Options
A.Smaller than α
B.Greater than α
C.Equal to α
D.Greater than or equal to α
Solution
Let f(x) = anxn + an-1xn-1 + ..... + a1x
f(0) = 0
f(α) = 0
∴ according to Rolle′s theorem
f′(x) = 0
has at least one root between (0, α)
⇒ f′(x) = 0 has a positive root less than α.
f(0) = 0
f(α) = 0
∴ according to Rolle′s theorem
f′(x) = 0
has at least one root between (0, α)
⇒ f′(x) = 0 has a positive root less than α.
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