Quadratic EquationHard
Question
If a + b + c = 0, then the quadratic equation 3ax2 + 2bx + c = 0has
Options
A.at least one root in (0,1)
B.one root in (2,3) and the other in (-2,-1)
C.imaginary roots
D.None of the above
Solution
Let f(x) = ax3 + bx2 + cx + d
∴ f (0) = d and f (1) = a + b + c + d = d (∵ a + b + c = 0)
∴ f(0) = f(1)
f is contnuous in the closed interval [0.1] and f is derivable in the open interval (0,1).
Also, f(0) = f(1)
∴ By Roll′s theorem,
f′(a) = 0 for 0 < a < 1
Now, f′(x)= 3ax3 + 2bx + c
⇒ f′(a) = 3aα2 + 2bα + c = 0
∴ Eq. (i) has exist at least one root in the interval (0,1).
Thus, f′(x) must have root in the interval (0,1). or 3ax2 + 2bx + c = 0 has root ∈ (0,1)
∴ f (0) = d and f (1) = a + b + c + d = d (∵ a + b + c = 0)
∴ f(0) = f(1)
f is contnuous in the closed interval [0.1] and f is derivable in the open interval (0,1).
Also, f(0) = f(1)
∴ By Roll′s theorem,
f′(a) = 0 for 0 < a < 1
Now, f′(x)= 3ax3 + 2bx + c
⇒ f′(a) = 3aα2 + 2bα + c = 0
∴ Eq. (i) has exist at least one root in the interval (0,1).
Thus, f′(x) must have root in the interval (0,1). or 3ax2 + 2bx + c = 0 has root ∈ (0,1)
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