Question

consider the function

$f(x)=3x^{2}+5x - 2$

a) determine the intervals on which $f$ is concave up and concave down.
$f$ is concave up on:
$f$ is concave down on:

b) based on your answer to part (a), determine the inflection points of $f$. each point should be entered as an ordered pair (that is, in the form $(x,y)$).
(separate multiple answers by commas.)

note: when using interval notation in webwork, remember that:
you use inf for $\infty$ and -inf for $-\infty$,
and use u for the union symbol.
enter dne if an answer does not exist.

note: you can earn partial credit on this problem.

Explanation:

Step1: Find the first derivative

Using the power rule \((x^n)^\prime = nx^{n - 1}\), for \(f(x)=3x^{2}+5x - 2\), \(f^\prime(x)=6x + 5\).

Step2: Find the second derivative

Differentiate \(f^\prime(x)\) again. \(f^{\prime\prime}(x)=(6x + 5)^\prime=6\).

Since \(f^{\prime\prime}(x)=6>0\) for all \(x\in(-\infty,\infty)\):

• For part (a), the function \(f(x)\) is concave - up on \((-\infty,\infty)\) and there is no interval where it is concave - down.
• For part (b), an inflection point occurs where \(f^{\prime\prime}(x)\) changes sign. Since \(f^{\prime\prime}(x)\) is a constant (\(f^{\prime\prime}(x) = 6\) never changes sign), there are no inflection points.

Answer:

a) \(f\) is concave up on: \((-\infty,\infty)\)
\(f\) is concave down on: DNE
b) DNE