Question

How do you find the maximum, minimum and inflection points and concavity for the function `f(x)=18x^3+5x^2-12x-17`?

Answer 1

Below

Explanation:

`f(x)=18x^3+5x^2-12x-17`

`f'(x)=54x^2+10x-12`

`f''(x)=108x+10`

For maximum or minimum points, `f'(x)=0`

`54x^2+10x-12=0`
`27x^2+5x-6=0`
`x=(-5+-sqrt(25+648))/54`
`x=(-5+-sqrt673)/54`
`x=(-5+sqrt673)/54` or `x=(-5-sqrt673)/54`

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To determine whether the point is maximum or minimum,
At `((-5+sqrt673)/54,-19.85)`,
`f''(x)=51.88>0`
Therefore, it is minimum and concave up at `((-5+sqrt673)/54,-19.85)`

At `((-5-sqrt673)/54,-0.57)`
`f''(x)=-51.88 <0`
Therefore, it is maximum and concave down at `((-5-sqrt673)/54,-0.57)`

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For inflexion points, `f''(x)=0`

`108x+10=0`
`x=-10/108=-5/54`

Test `(-5/54,-15.86)`
`f''(0)=0+10=10>0`
`f''(-5/54)=0`
`f''(-1/2)=-44 <0`
Therefore, since there is a change in concavity, a point of inflexion occurs at `(-5/54,-15.86)`