Question

(a) For f(x) = 1 4 x 4 − 6x 2 find the intervals where f(x) is concave up, and the intervals where f(x) is concave down, and the inflection 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 a polynomial. Conclude that f 00(x) is either always positive or always negative on each of the intervals (−∞, −2), (−2, 2), and (2, ∞).

iv. Evaluate f 00(c) at one point c in each of these three intervals to see if f 00(c) > 0 or f 00(c) < 0, and use this computation to indicate where f(x) is concave up and where f(x) is concave 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, and all inflection points of g(x). HINT: Use same steps as previous problem

Answer 1