Question

determine the intervals on which the following function is concave up or concave down. identify any inflection points.
( f(x)=2 x^{4}+40 x^{3}+300 x^{2}-x - 3 )
determine the intervals on which the following function is concave up or concave down select the correct choice and, if necessary, fill in the answer box(es) to complete your choice
a. the function is concave up on and concave down on
(simplify your answers type your answers in interval notation. use a comma to separate answers as needed.)
b. the function is concave up on and the function is never concave down
(simplify your answer type your answer in interval notation. use a comma to separate answers as needed.)
c. the function is concave down on and the function is never concave up
(simplify your answer. type your answer in interval notation use a comma to separate answers as needed.)
d. the function is never concave up nor concave down

Explanation:

Step1: Find the first derivative

Using the power rule \((x^n)^\prime = nx^{n - 1}\), for \(y = f(x)=2x^{4}+40x^{3}+300x^{2}-x - 3\), we have \(f^\prime(x)=8x^{3}+120x^{2}+600x-1\).

Step2: Find the second derivative

Differentiate \(f^\prime(x)\) again. \(f^{\prime\prime}(x)=24x^{2}+240x + 600\).
Factor out \(24\): \(f^{\prime\prime}(x)=24(x^{2}+10x + 25)\).
Using the perfect - square formula \(a^{2}+2ab + b^{2}=(a + b)^{2}\), we get \(f^{\prime\prime}(x)=24(x + 5)^{2}\).

Step3: Analyze the sign of the second derivative

Since \((x + 5)^{2}\geq0\) for all real \(x\) and \(24>0\), then \(f^{\prime\prime}(x)=24(x + 5)^{2}\geq0\) for all \(x\in(-\infty,\infty)\).
The equality holds when \(x=-5\) (at this point, the concavity does not change).

Answer:

B. The function is concave up on \((-\infty,\infty)\) and the function is never concave down.