let ( f(x)=x^{4}+6x^{3}+48x^{2}). a. find the inflection points. enter your answers as ordered pairs. b…

let ( f(x)=x^{4}+6x^{3}+48x^{2}). a. find the inflection points. enter your answers as ordered pairs. b. identify interval(s) where the graph of ( f ) is concave up. write none if there is no answer. c. identify interval(s) where the graph of ( f ) is concave down. write none if there is no answer.

let ( f(x)=x^{4}+6x^{3}+48x^{2}). a. find the inflection points. enter your answers as ordered pairs. b. identify interval(s) where the graph of ( f ) is concave up. write none if there is no answer. c. identify interval(s) where the graph of ( f ) is concave down. write none if there is no answer.

Answer

Explanation:

Step1: Find the first - derivative

Given (f(x)=x^{4}+6x^{3}+48x^{2}). Using the power rule ((x^n)^\prime = nx^{n - 1}), we have (f^\prime(x)=4x^{3}+18x^{2}+96x).

Step2: Find the second - derivative

Differentiate (f^\prime(x)) again. (f^{\prime\prime}(x)=12x^{2}+36x + 96=12(x^{2}+3x + 8)).

Step3: Find inflection points

Set (f^{\prime\prime}(x) = 0). For the quadratic equation (x^{2}+3x + 8=0), the discriminant (\Delta=b^{2}-4ac) where (a = 1), (b = 3), and (c = 8). (\Delta=3^{2}-4\times1\times8=9 - 32=- 23<0). So, there are no real - valued inflection points.

Step4: Determine concavity

Since (f^{\prime\prime}(x)=12(x^{2}+3x + 8)) and the coefficient of (x^{2}) in (x^{2}+3x + 8) is positive ((a = 1>0)) and (\Delta<0), (f^{\prime\prime}(x)>0) for all real (x).

Answer:

a. NONE b. ((-\infty,\infty)) c. NONE