Application of DerivativeMedium
Question
If 27a + 9b + 3c + d = 0, then the equation 4ax3 + 3bx2 + 2cx + d = 0, has at least one real root lying between-
Options
A.0 and 1
B.1 and 3
C.0 and 3
D.None
Solution
27a + 9b + 3c + d = 0
f′(x) = 4ax3 + 3bx2 + 2cx + d = 0
f(x) = ax4 + bx3 + cx2 + dx + e
f(0) = e
f(3) = 81a + b × 27 + 9c + 3d + e
3(27a + 9b + 3c + d) + e
3 × (0) + e = e
f(0) = f(3)
Rolle′s proved
intval (0, 3)
f′(x) = 4ax3 + 3bx2 + 2cx + d = 0
f(x) = ax4 + bx3 + cx2 + dx + e
f(0) = e
f(3) = 81a + b × 27 + 9c + 3d + e
3(27a + 9b + 3c + d) + e
3 × (0) + e = e
f(0) = f(3)
Rolle′s proved
intval (0, 3)
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