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https://gateoverflow.in/423/gate2008-25

Why we have not done 3rd derivative test..3rd derivative will give different results 

in Calculus by (34 points) | 20 views
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How will it give different result?
It will give same result as the selected ans, isnot it?
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f′′(x)=36x2−96x+48

at x=0,f′′(x)=48>0 it means that  x=0 is local minima.

but at x=2,f′′(x)=0 so we can't apply second derivative test.

So, we can apply third derivative test

3rd derivative = 72x- 96

At x=2, 3rd derivative >0 therefore local maxima? But ans do have 2 as local maxima.

What i am doing wrong?
0
No. If $f^{'''}(x_0)\neq0$ then we get neither minima nor maxima at the point $x=x_0$
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I had read somewhere that

For even k, if f^k(x) >0 then local minima at x.

For odd k, if f^k(x) >0 then local maxima at x.

f^k(x) is k the derivative of f(x).

What i am thinking wrong?

Please provide some resource to clear this.

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