Problem

(a) Find the intervals on which $f$ is increasing…

04:12

Problem 2 Medium Difficulty

(a) Find the intervals on which $f$ is increasing or decreasing.
(b) Find the local maximum and minimum values of $f .$
(c) Find the intervals of concavity and the inflection points.
$f(x)=4 x^{3}+3 x^{2}-6 x+1$

Answer

A. Increasing in $(-\infty,-1) \cup\left(\frac{1}{2}, \infty\right)$
Decreasing in $\left(-1, \frac{1}{2}\right)$
B. The local maximum is $f(-1)=6,$ so $(-1,6)$
The local minimum is $f\left(\frac{1}{2}\right)=-\frac{3}{4},$ so $\left(\frac{1}{2},-\frac{3}{4}\right)$
C. Inflection point at $x=-\frac{1}{4}$
Concave downward $\left(-\infty,-\frac{1}{4}\right)$
Concave upward in $\left(-\frac{1}{4}, \infty\right)$

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Video Transcript

for this question. We have three different parts. So let's first right out of the first of the narrative and the second that curative. So we were right out the front of the irritate. We can just do the baby affect her, factoring into something like this which directly gives us the roots off this first of the theater. And for the second of their two, we have did you for explosives. So probably we want to find the increasing decrease interval. So we just let if prime off X equals zero. So we have exit post minus one. The exit cross will have, which means we have three step intervals from negative infinity to negative one thrown negative 1 to 1/2 for on one have to pause appear taking unity. So for the first interval, if prime of X is positive, the muse f is increasing on the second terrible If prime affects this connective so every is decreasing. On the last interval the first of the narrative eight supportive. So it's increasing. I face increasing again and the footprint leave property is directly related to party. We have a local necks. We have a local maximum at X equals two minus one because, um, when X equals two minus one the function on the left the function is increasing and the on the right the function istea crazy. So it's ah, peak. Honestly, value will be a minus one. There's six and for the same reason we have a local Minuteman that X equals to one has so a 4 4.5 equals two minus three or four. And for the concave, it ease way we look at the second the narrative and it just let the second narrative because it's euro the solution will g minus 1/4. Uh, since we only have one solution, we have to stop being talk of us from next If he obviously tonight give wonderful on the phone connective 1/4 to positive unity. So over the first interval, the second of narrative, it's negative. The music conclave hung worked and the over the second terrible on the second of curative is positive. That means it's the collectivity is upward and this is the conservatives are different around X equals to one or four. So we have a reflection point. But but which? Um yes at X equals two minus wonderful

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