Find the turning points of the following functions of x and also determine their nature:-   Y=(3X-5)3(3X-7)please could you work it step by steps. thanks

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find derivative of the function `y = (3x-5)^3*(3x-7). ` Check if derivative changes its sign at a point. If the derivative changes its sign at that point, then you have found a turning point.

Use the product rule, power rule and chain rule to find the derivative.

`dy/dx = 3(3x-5)^2*(3x-7) + 3(3x-5)^3`

Expand the binomials:

`dy/dx = 3(9x^2 - 30x + 25)(3x-7) + 3(27x^3 - 135x^2 + 225x - 125)`

Opening the brackets yields:

`dy/dx = 81x^3 - 189x^2 - 270x^2 + 630x + 225x - 525 + 81x^3 - 405x^2 + 675x - 375`

`` `dy/dx = 162x^3 - 864x^2 + 1530x - 900`

`dy/dx = 9(18x^3 - 96x^2 + 170x - 100)`

Put `dy/dx = 0`  => `18x^3 - 96x^2 + 170x - 100 =`  0

You should find the roots of the equation `18x^3 - 96x^2 + 170x - 100 = 0`  among the divisors of 100/18.

The roots of `18x^3 - 96x^2 + 170x - 100 = 0`  become the x coordinates of the turning points of the function.

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