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Q #9. Consider the equation \(f(x) = x^4 - 32x^2 + 8\) 1. Find the interval on which \(f\) is increasing. 2. Find the interval on which \(f\) is decreasing. 3. Find the inflection points. 4. Find the interval on which \(f\) is concave up. 5. Find the interval on which \(f\) is concave down. 

          Q #9. Consider the equation
\(f(x) = x^4 - 32x^2 + 8\)
1. Find the interval on which \(f\) is increasing.
2. Find the interval on which \(f\) is decreasing.
3. Find the inflection points.
4. Find the interval on which \(f\) is concave up.
5. Find the interval on which \(f\) is concave down.
Show more…
Q #9. Consider the equation f(x) = x^4 - 32x^2 + 8 1. Find the interval on which f is increasing. 2. Find the interval on which f is decreasing. 3. Find the inflection points. 4. Find the interval on which f is concave up. 5. Find the interval on which f is concave down.

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Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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Q #9. Consider the equation f(x) = x^4 - 32x^2 + 8 1. Find the interval on which f is increasing. 2. Find the interval on which f is decreasing. 3. Find the inflection points. 4. Find the interval on which f is concave up. 5. Find the interval on which f is concave down.
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Transcript

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00:01 Okay, let's start with the first derivative.
00:03 4x cubed minus 64x is equal to zero.
00:09 Take 4x out.
00:11 X squared minus 64.
00:17 Wait a second, that's not 64, 16.
00:20 This is 16.
00:24 So x equals zero, x equals four, and x equals negative four are the critical points.
00:34 Negative infinity f prime.
00:37 So negative four and zero and four and infinity.
00:51 Okay, so let's put negative five for the first derivative.
00:56 You get 25 minus 9 positive negative and then put negative 2.
01:03 Negative, negative, positive.
01:08 What about 1? positive, negative, 5, positive, positive.
01:13 So decreasing, increasing, decreasing, increasing.
01:18 Okay, increasing from negative four to zero.
01:29 Union four to infinity and decreasing from negative infinity to negative four.
01:40 Union zero to four.
01:46 And then here is a local min, local max, and then the local min.
01:52 So local min, minimum, at plus or minus 4 comma negative 248.
02:04 And local max at 0 comma 8.
02:10 Okay, second derivative.
02:14 Second derivative is 12x squared minus 64...
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