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Monotony of the functionVerify the monotony of the function y=x^3-x^2+x+e^x.

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colorcolour | Student, College Freshman

Posted May 18, 2011 at 1:20 AM via web

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Monotony of the function

Verify the monotony of the function y=x^3-x^2+x+e^x.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted May 9, 2013 at 6:03 PM (Answer #1)

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You need to use monotonicity theorem, hence, you need to find derivative of the function, such that:

`f'(x) = 3x^2 - 2x + 1 + e^x`

You need to find the second order derivative of the function, such that:

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

You need to find the third order derivative of the function, such that:

`f'''(x) = 6 + e^x`

Since `f'''(x) > 0` for all x in R, hence, `f''(x),f'(x) > 0.`

Hence, testing the monotony of the function, yields that `f(x) = x^3-x^2+x+e^x` increases if `x in R` .

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