For function f ( x ) = x 4 - 2 x 3 + 3 with domain ( -infinity, infinity ) , find the...

Question:

For function {eq}f(x)=x^4-2x^3+3 {/eq} with domain {eq}(-\infty,\infty) {/eq}, find the following:

a. Any local extrema

b. Any inflection points

Local Extrema:

The local extrema of a function {eq}f(x) {/eq} are obtained setting the first derivative to zero, i e.

{eq}f'(x)=0. {/eq}

The inflection points of {eq}f(x) {/eq} are obtained setting the second derivative to zero, i e.

{eq}f''(x)=0. {/eq}

Answer and Explanation:

Given the function

{eq}f(x)=x^4-2x^3+3 {/eq}

its first and second derivatives are found as

{eq}f'(x)=4x^3-6x^2=2x^2(2x-3) \\ f''(x)= 12x^2 -12x= 12x(x-1). {/eq}

a. The local extrema of the function are found setting the first derivative to zer

{eq}f'(x)=0 \Rightarrow 2x^2(2x-3) =0 \\ \displaystyle \Rightarrow x=0, \; x= \frac{3}{2}. {/eq}

b. The inflection points are found setting the second derivative to zero

{eq}f''(x)=0 \Rightarrow 12x(x-1)=0 \\ \Rightarrow x=0,\; x=1. {/eq}


Learn more about this topic:

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Understanding Concavity and Inflection Points with Differentiation

from Math 104: Calculus

Chapter 10 / Lesson 6
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