Mathematics: Megan hiked 15.12 miles in 6.3 hours is meghan hike the same number of miles each hour how many miles did she hike each hour

Relative maximum at x=12 Relative minimum at x=-4 Inflection point at x=4Step-by-step explanation:To find all the critical points of the function we need the first derivative of the functionAccording to Fermat s theorem, it will be a critical point in y (x)=0, thenWith the test of the second derivative, we can know if the points are relative extrema or inflections.The second derivative test says: if then x is a relative maximum, if then x is a relative minimum. So x=12 is a relative maximum and...

Megan hiked 15.12 miles in 6.3 hours is meghan hike

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uberagentkenny

16.05.2020

Mathematics

Relative maximum at x=12

Relative minimum at x=-4

Inflection point at x=4

Step-by-step explanation:

To find all the critical points of the function we need the first derivative of the function

$y(x)=-2x^3+24x^2+288x-35\\y'(x)=-2(3)x^{3-1}+24(2)x^{2-1}+288(1)\\y'(x)=-6x^2+48x+288$

According to Fermat's theorem, it will be a critical point in y'(x)=0, then

$y'(x)=-6x^2+48x+288\\0=-6x^2+48x+288\\\\-6x^2+48x+288=0\\-6(x^2-8x-48=0\\x^2-8x-48=0\\(x-12)(x+4)=0\\x_1=12\\x_2=-4\\$

With the test of the second derivative, we can know if the points are relative extrema or inflections.

$y'(x)=-6x^2+48x+288\\y''(x)=-6(2)x^{2-1}+48(1)\\y''(x)=-12x+48$ $y''(12)=-12(12)+48\\y''(12)=-144+48=-96$

$y''(-4)=-12(-4)+48\\y''(-4)=48+48=96$

The second derivative test says: if f''(x) then x is a relative maximum, if f''(x)0 then x is a relative minimum. So x=12 is a relative maximum and x=-4 is a relative minimum.

To find the inflection point y''(x)=0

$y''(x)=-12x+48\\0=-12x+48\\12x=48\\x=48/12=4$

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Megan hiked 15.12 miles in 6.3 hours is meghan hike the same number of miles each hour how many miles did she hike each hour

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