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  3. (a) for f(x) = 1 4 x 4 − 6x 2 find the inter

(a) For f(x) = 1 4 x 4 − 6x 2 find the intervals where...

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Basic Math

(a) For f(x) = 1 4 x 4 − 6x 2 find the intervals where f(x) isconcave up, and the intervals where f(x) is concave down, and theinflection points of f(x) by the following steps:

i. Compute f 0 (x) and f 00(x).

ii. Show that f 00(x) is equal to 0 only at x = −2 and x =2.

iii. Observe that f 00(x) is a continuous since it is apolynomial. Conclude that f 00(x) is either always positive oralways negative on each of the intervals (−∞, −2), (−2, 2), and (2,∞).

iv. Evaluate f 00(c) at one point c in each of these threeintervals to see if f 00(c) > 0 or f 00(c) < 0, and use thiscomputation to indicate where f(x) is concave up and where f(x) isconcave down.

v. Indicate the inflection points (c, f(c)) where f 00(x)changes sign at x = c.

(b) For g(x) = 5 6 x 3 − 1 12x 4 find the intervals where g(x)is concave up, and the intervals where g(x) is concave down, andall inflection points of g(x). HINT: Use same steps as previousproblem

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