Name: Solving Algebraic Inequalities
Uploaded: 2019-07-19
Duration: 6 min 24 s
Description: Do you know, like algebraic equations, inequalities are solved to find the solution? In this video, we will learn the method of solving algebraic inequalities in detail.

Question

Let

f (x) = x^{3} + 3 x^{2} + 9 x + 6 \sin x,

then find the number of real roots of the equation

\frac{1}{x - f (1)} + \frac{2}{x - f (2)} + \frac{3}{x - f (3)} = 0

Accepted Answer

Given $f (x) = x^{3} + 3 x^{2} + 9 x + 6 \sin x$

Now,

$f (0) = 0$

Now differentiate $f (x)$ w.r.t $x$

$f^{'} (x) = 3 x^{2} + 6 x + 9 + 6 \cos x = 3 {(x + 1)}^{2} + 6 (1 + \cos x)$

$∴ f^{'} (x) > 0$

$∴ f (x)$ is monotonically increasing graph.

$∴ f (3) > f (2) > f (1) > f (0)$

$∴ f (3) > f (2) > f (1) > 0 \{∵ if x_{1} > x_{2} \Rightarrow f (x_{1}) > f (x_{2})\}$

Let $f (1) = a, f (2) = 9, f (3) = c$

Let $g (x) = \frac{1}{x - f (1)} + \frac{2}{x - f (2)} + \frac{3}{x - f (3)} = 0$

$\Rightarrow (x - b) (x - c) + 2 (x - a) (x - c) + 3 (x - b) (x - a) = 0 (x \neq a, b, c)$

$\Rightarrow 3 x^{2} - (b + c + 2 a + 2 c + 3 b + 3 a) x + (b c + 2 a c + 3 a b) = 0$

$\Rightarrow 3 x^{2} - (5 a + 4 b + 3 c) x + (3 a b + 2 a c + b c) = 0$

Now, discriminant of this equation

$△ = {(5 a + 4 b + 3 c)}^{2} - 4 (3) (3 a b + b c + 2 a c)$

$\Rightarrow △ = 25 a^{2} + 16 b^{2} + a c^{2} + 4 a b + 6 a c + 12 b c$

$∴ △ > 0 (∵ a > b > c > 0)$

$∴ g (x)$ has two distinct real roots.

Let $f (x) = x^{3} + 3 x^{2} + 9 x + 6 \sin x,$ then find the number of real roots of the equation $\frac{1}{x - f (1)} + \frac{2}{x - f (2)} + \frac{3}{x - f (3)} = 0$

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

Let f(x)=x3+3x2+9x+6sinx, then find the number of real roots of the equation 1x-f(1)+2x-f(2)+3x-f(3)=0

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

Let $f (x) = x^{3} + 3 x^{2} + 9 x + 6 \sin x,$ then find the number of real roots of the equation $\frac{1}{x - f (1)} + \frac{2}{x - f (2)} + \frac{3}{x - f (3)} = 0$