The number of real roots of the equation $e^{4x}+2e^{3+}-e^x-6=0$  is:

Correct Option - 3


Detailed Solution:

$e^{4x}+2e^{3x}-e^x-6=0$

$e^x=t\in \left(0,\infty \right)$

$t^4+2t^3-t-6=0$

Let f (t) = t⁴ + 2t³ – t – 6

f’ (t) = 4t³ + 6t² – 1f

Þf’ (0) = -1, f’ (+ $\infty )$ = + $\infty$

For t > 0

Þ f’ (t) = 0 has only one root.

One solution of f (t) = 0 is possible

<p><span contenteditable="false"> <math> <mrow> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mn>4</mn> <mi>x</mi> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mn>3</mn> <mi>x</mi> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msup> <mo>−</mo> <mn>6</mn> <mo>=</mo> <mn>0</mn> </mrow> </math> </span></p><p><span contenteditable="false"> <math> <mrow> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>t</mi> <mo>∈</mo> <mrow> <mo> (</mo> <mrow> <mn>0</mn> <mo>, </mo> <mo>∞</mo> </mrow> <mo>)</mo> </mrow> </mrow> </math> </span></p><p><span contenteditable="false"> <math> <mrow> <msup> <mrow> <mi>t</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mrow> <mi>t</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mi>t</mi> <mo>−</mo> <mn>6</mn> <mo>=</mo> <mn>0</mn> </mrow> </math> </span></p><p>Let f (t) = t⁴ + 2t³ – t – 6</p><div><div><picture><source srcset="" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1754470729phpSpZCwa.jpeg" alt="" width="184" height="161" loading="lazy"></picture></div></div><p>f’ (t) = 4t³ + 6t² – 1f</p><p>Þf’ (0) = -1, f’ (+<span contenteditable="false"> <math> <mrow> <mo>∞</mo> <mo stretchy="false">)</mo> </mrow> </math> </span> &lt;!- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1815996415"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;= + <span contenteditable="false"> <math> <mrow> <mo>∞</mo> </mrow> </math> </span>&lt;!- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1815996416"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;</p><p>For t &gt; 0</p><p>Þ f’ (t) = 0 has only one root.</p><p>One solution of f (t) = 0 is possible</p>

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