SATHEE: Application of Derivatives 4 Question 63

Application of Derivatives 4 Question 63

66. The number of distinct real roots of $x^{4} - 4 x^{3} + 12 x^{2} + x - 1 = 0$ is……

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Solution:

$f (x) = x^{4} - 4 x^{3} + 12 x^{2} + x - 1$

$f^{'} (x) = 4 x^{3} - 12 x^{2} + 24 x + 1$

$f^{''} (x) = 12 x^{2} - 24 x + 24 = 12 (x^{2} - 2 x + 2)$

$= 12 (x - 1)^{2} + 1 > 0 \forall x$

$\Rightarrow f^{'} (x)$ is increasing.

Since, $f^{'} (x)$ is cubic and increasing.

$\Rightarrow f^{'} (x)$ has only one real root and two imaginary roots.

$∴ f (x)$ cannot have all distinct roots.

$\Rightarrow$ Atmost 2 real roots.

Now, $f (- 1) = 15, f (0) = - 1, f (1) = 9$

$∴ f (x)$ must have one root in $(- 1, 0)$ and other in $(0, 1)$ .

$\Rightarrow 2$ real roots.

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Application of Derivatives 4 Question 63

66. The number of distinct real roots of x4−4x3+12x2+x−1=0 is……

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66. The number of distinct real roots of $x^{4} - 4 x^{3} + 12 x^{2} + x - 1 = 0$ is……