How do you find all local maximum and minimum points using the second derivative test given $y = x^{5} - x$ $y=x^5-x$ ?

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Answer 1 · 2017-10-23T02:00:38

$y_max = y(approx-0.669) approx 0.535$
$y_min = y(approx+0.669) approx -0.535$

$y = x^5-x$

$dy/dx = 5x^4-1$

$(d^2y)/dx^2 = 20x$

For local extrema $dy/dx=0$

$:. 5x^4-1 =0$

$5x^4 =1$

$x^4 =1/5$

$x = +-(1/5)^(1/4), +-(1/5)^(1/4)i$

Since we are only interested in real roots

$x=+-(1/5)^(1/4) approx +-0.669$

For a local maximum $(d^2y)/dx^2 <0$

For a local minimum $(d^2y)/dx^2 >0$

Testing our results from above

$xapprox+0.669 -> 20xx x >0 -> y(approx+0.669) = y_min$

$xapprox-0.669 -> 20xx x <0 -> y(approx-0.669) = y_max$

Hence:

$y_max = y(approx-0.669) approx 0.535$
$y_min = y(approx+0.669) approx -0.535$

We can see the local maximum and minimum on the graph of $y$ below.

graph{x^5-x [-2.434, 2.434, -1.217, 1.217]}

How do you find all local maximum and minimum points using the second derivative test given y=x^5-xy=x5−xy=x^5-x?