how to Use the graph of f 'and f'' to estimate the maximum and minimum points. the  polynomials equation f(x)= 2x^6 + 3x^5 + 3x^3 - 2x^2.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember that you may find the maximum and minimum points of the function solving the equation `f'(x) = 0`  such that:

`f'(x) = (2x^6 + 3x^5 + 3x^3 - 2x^2)'`

`f'(x) = (12x^5 + 15x^4 + 9x^2 - 4x)`

You need to solve the equation `f'(x) = 0`  such that:

`12x^5 + 15x^4 + 9x^2 - 4x = 0`

You need to factor out x such that:

`x(12x^4 + 15x^3 + 9x - 4) = 0`

`x = 0`

`12x^4 + 15x^3 + 9x - 4 = 0 `

You need to write the factored form of polynomial `12x^4 + 15x^3 + 9x - 4`  such that:

`12x^4 + 15x^3 + 9x - 4 = (x + 1.616)(x - 0.352)(0.007 + 0.766i)(0.007- 0.766i)`

Hence, the function reaches its extreme points at `x = -1.616,x = 0, x = 0.352` .

Notice that the function reches its minimum at x = -1.616.

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meepoh90 | Student, Undergraduate | (Level 1) eNoter

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how to get  12x^4+15x^3+9x-4=(x+1.616)(x-0.352)(0.007+0.766i)(0.007-0.766)? is it any formulae to calculate it? how about f''(x)=?

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