The number of distinct real roots of x4 – 4x + 1 = 0 is:

Correct Option - 2


Detailed Solution:

$f\left(x\right)=x^4-4x+1=0$

$f\text{'}\left(x\right)=4x^3-4$

$=4\left(x-1\right)\left(x^2+1+x\right)$

$\Rightarrow$ Two solution

<p><span contenteditable="false"> <math> <mrow> <mi>f</mi> <mrow> <mo> (</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mrow> </math> </span></p><div><div><picture><source srcset="" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1755575068phpUoIkud.jpeg" alt="" width="176" height="111" loading="lazy"></picture></div></div><p><span contenteditable="false"> <math> <mrow> <mi>f</mi> <mo>'</mo> <mrow> <mo> (</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> </mrow> </math> </span></p><p><span contenteditable="false"> <math> <mrow> <mo>=</mo> <mn>4</mn> <mrow> <mo> (</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo> (</mo> <mrow> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> </span></p><p><span contenteditable="false"> <math> <mrow> <mo>⇒</mo> </mrow> </math> </span> Two solution</p>

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