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JEE MainMathematicsModerateTheory of Equations

The graphs of y = x^4 - 2x^2 + 5 is

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y=x4−2x2+5y = x^{4} - 2x^{2} + 5
y′=4x3−4x+0y^{'} = 4x^{3} - 4x + 0
⇒4n(n2−1)=0\Rightarrow 4n\left( n^{2} - 1 \right) = 0
⇒x=0,±1\Rightarrow x = 0, \pm 1 (Critical points)
y"=12x2−4y" = 12x^{2} - 4
y"(0)=−4<0y"(0) = - 4 < 0 (maximum at x = 0)
y"(1)=8>0y"(1) = 8 > 0 (maximum at x = 1)
y"(−1)=8>0y"( - 1) = 8 > 0 (minimum at x = - 1)

Source: Grass-G / M-06 Mathematics Paper 14 October.docx

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