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Ex 2.6
9. Find the stationary points on the curve y= (4x²-1)^4 and determine their nature


Sagot :

[tex]y=(4x^2-1)^4\\ y'=4(4x^2-1)^3\cdot8x=32x(4x^2-1)^3\\\\ 32x(4x^2-1)^3=0\\ x=0 \vee 4x^2-1=0\\ 4x^2-1=0\\ 4x^2=1\\ x^2=\dfrac{1}{4}\\ x=-\dfrac{1}{2} \vee x=\dfrac{1}{2}\\ x\in\left\{-\dfrac{1}{2},0,\dfrac{1}{2}\right\}\leftarrow \text{stationary points}\\\\ [/tex]

[tex]y''=32((4x^2-1)^3+x\cdot3(4x^2-1)^2\cdot8x)\\ y''=32(4x^2-1)^2(4x^2-1+24x^2)\\ y''=32(4x^2-1)^2(28x^2-1)\\\\ y''(0)=32(4\cdot0^2-1)^2(28\cdot0^2-1)=32\cdot(-1)^2\cdot(-1)=-32[/tex]
[tex]-32<0 \Rightarrow[/tex] there's a (local) maximum at [tex]x=0[/tex].

For [tex]x\in\left\{-\dfrac{1}{2},\dfrac{1}{2}\right\}[/tex], [tex]y''=0[/tex] because the factor [tex](4x^2-1)^2=0[/tex].
You have to check values of [tex]y'[/tex] around [tex]x\in\left\{-\dfrac{1}{2},\dfrac{1}{2}\right\}[/tex].

[tex]\forall x\in(\infty,-\frac{1}{2})\, y'<0 \Rightarrow y\searrow\\ \forall x\in(-\frac{1}{2},0)\, y'>0 \Rightarrow y\nearrow\\\\ \forall x\in(0,\frac{1}{2})\, y'<0 \Rightarrow y\searrow\\ \forall x\in(\frac{1}{2},\infty)\, y'>0 \Rightarrow y\nearrow [/tex]

The above means that at [tex]x=-\dfrac{1}{2}[/tex] and [tex]x=-\dfrac{1}{2}[/tex] there are (global) maxima.