First slide

Theory of equations

Question

$If the graph of f (x) = 2 x^{3} + a x^{2} + b x (a, b \in N) cuts the x -axis at three distinct points, then minimum value of (a + b) is$

Moderate

A

2
B

3
C

4
D

5

Solution

$\begin{aligned} f (x) = 2 x^{3} + a x^{2} + b x; a, b \in N \\ = x (2 x^{2} + a x + b) \end{aligned}$
$Given f (x) cuts x -axis at 3 distinct points then$
$2 x^{2} + a x + b must have 2 distinct real roots$
$\begin{aligned} ∴ a^{2} - 8 b > 0 \\ \Rightarrow a^{2} > 8 b \end{aligned}$
$Since a, b \in N, minimum value of a + b is 4 when a = 3 and b=1$

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