Question

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.

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