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

Oct 23, 2017

${y}_{\max} = y \left(\approx - 0.669\right) \approx 0.535$
${y}_{\min} = y \left(\approx + 0.669\right) \approx - 0.535$

#### Explanation:

$y = {x}^{5} - x$

Apply power rule

$\frac{\mathrm{dy}}{\mathrm{dx}} = 5 {x}^{4} - 1$

$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = 20 x$

For local extrema $\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

$\therefore 5 {x}^{4} - 1 = 0$

$5 {x}^{4} = 1$

${x}^{4} = \frac{1}{5}$

$x = \pm {\left(\frac{1}{5}\right)}^{\frac{1}{4}} , \pm {\left(\frac{1}{5}\right)}^{\frac{1}{4}} i$

Since we are only interested in real roots

$x = \pm {\left(\frac{1}{5}\right)}^{\frac{1}{4}} \approx \pm 0.669$

For a local maximum $\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 < 0$

For a local minimum $\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 > 0$

Testing our results from above

$x \approx + 0.669 \to 20 \times x > 0 \to y \left(\approx + 0.669\right) = {y}_{\min}$

$x \approx - 0.669 \to 20 \times x < 0 \to y \left(\approx - 0.669\right) = {y}_{\max}$

Hence:

${y}_{\max} = y \left(\approx - 0.669\right) \approx 0.535$
${y}_{\min} = y \left(\approx + 0.669\right) \approx - 0.535$

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]}