1. ## Linearization question

Find linearization of f (x)= x^(1/3) With center a = 27

I got 1/27*x +2

But then it asks: is this linearization greater than or smaller than f (x) when x is close to but not equal to 27?

What does this question mean. I'm not sure what it is asking and how.

2. ## Re: Linearization question

let $f(x) = x^{1/3},~l(x)=\dfrac{x}{27}+2$

let $\epsilon > 0$

$f(27 +\epsilon) \overset{?}{\lessgtr} l(27 + \epsilon)$

$f(27 -\epsilon) \overset{?}{\lessgtr} l(27 -\epsilon)$

3. ## Re: Linearization question

What it is asking is, "if x is close to 3, is $\displaystyle x^{1/3}$ greater than or less than $\displaystyle \left(\frac{1}{27}\right)x+ 2$?" Equivalently, "is $\displaystyle x^{1/3}- \left(\frac{1}{27}\right)x- 2$ positive or negative?". Of course, one way of getting the "linearization" of $\displaystyle x^{1/3}$ around x= 27 is to take the first two terms, $\displaystyle (x- 27)^0$ and [tex](x- 27)^1[/ex], of the Taylor's series about x= 3. What is the coefficient of the quadratic, $\displaystyle (x- 27)^2$ term? Is it positive or negative?

4. ## Re: Linearization question

Wait, i'm kinda confused. Does this involve summation/integral? Because we didn't learn that yet, and it was part of the review exam which my professor said wouldn't be on the actual exam (summation)

5. ## Re: Linearization question

But then it asks: is this linearization greater than or smaller than f (x) when x is close to but not equal to 27?

What does this question mean. I'm not sure what it is asking and how.
If a function is concave up at the point of tangency, then points on a tangent line close to the point of tangency will be below points on the function curve.

If a function is concave down at the point of tangency, then points on the tangent line close to the point of tangency will be above points on the function curve.

$f(x) = x^{1/3} \implies f'(x) = \dfrac{1}{3x^{2/3}} \implies f''(x) = -\dfrac{2}{9x^{5/3}}$

note $f''(27) < 0 \implies f(x)$ is concave down, therefore the tangent line, $L(x)$, used to determine a linear approximation lies above the curve at values of $x$ close to $x=27$ ... $L(x) > f(x)$

6. ## Re: Linearization question

Thank YOU! This makes much more sense !!