First slide

Mean value theorems

Question

$Let a, b, c, d are non - zero real numbers such that 6 a + 4 b + 3 c + 3 d = 0, then the equation$ $a x^{3} + b x^{2} + c x + d = 0 has$

Moderate

A

$at least are root in [- 2, 0]$
B

$at least one root in (0, 2)$
C

$at least one root in [- 2, 2]$
D

$no root in [- 2, 2]$

Solution

$\begin{array}{l} We have a x^{3} + b x^{2} + c x + d = 0 \\ Let f (x) = \frac{a x^{4}}{4} + \frac{b x^{3}}{3} + \frac{c x^{2}}{2} + d x + e \\ ∴ f (0) = e \\ f (2) = 4 a + \frac{8 b}{3} + 2 c + 2 d + e = \frac{(12 a + 8 b + 6 c + 6 d)}{3} + e \\ = \frac{2}{3} (6 a + 4 b + 3 c + 3 d) + e = 0 \Rightarrow f (2) = e \\ ∴ By Rolle's theorem, there exist at least one value of \\ x \in (0, 2) such that f^{'} (x) = 0 \end{array}$

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