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Let g be the function give by g(x) = x^4 - 4x^3 + 6x^2 - 4x + k where k is constant. A.On what...

Question:

Let g be the function give by {eq}g(x) = x^4 - 4x^3 + 6x^2 - 4x + k {/eq} where k is constant.

A.On what intervals is g increasing? Justify your answer.

B.On what interval is g concave upward? Justify your answer.

C.Find the value of k for which g has 5 as its relative minimum. Justify your answer.

Monotony and concavity:

We are looking for the monotony of this function, so we will need to differentiate the function with respect to the variable 'x'.

Then, we need the concavity, so we will need the second derivative.

Note that in the interval(s) where the first derivative is positive, the function increases, while in the interval(s) where the derivative is negative, the funciton decreases.

Answer and Explanation:

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First of all, the given function is,

{eq}g(x) = x^4 - 4x^3 + 6x^2 - 4x + k {/eq}

The derivative is,

{eq}g'(x) = 4x^3-12x^2+12x-4 {/eq}

We are...

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Concavity and Inflection Points on Graphs

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Chapter 9 / Lesson 5
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