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  4. The number of real solutions of x7+5 x3+3 x+1= 0 is equal to.

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Q. The number of real solutions of $x^{7}+5 x^{3}+3 x+1= 0$ is equal to_____.

JEE MainJEE Main 2022Application of Derivatives

Solution:

$f(x)=x^{7}+5 x^{3}+3 x+1$
$f^{\prime}(x)=7 x^{6}+15 x^{2}+3 > 0$
$\therefore f ( x )$ is strictly increasing function
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$\therefore$ no. of real solution $= 1$