Question

How to analytically find the intervals on which the function y = 2x^4 - 4x^2 + 1 is
1)increasing
2)decreasing
3)concave up
4)concave down

Also find any local extreme values and inflection points.

Thanks.

(1) The function is increasing when the first derivative dy/dx = 8x^3 - 8x is positive
(2) The function is decreasing there the same derivative is negative.
(3) The function is concave up where the second derivative, 24 x^2 - 8, is positive.
(4) The function is concave down where the second derivative is negative.
Extreme values occur whereever dy/dx = 0
Inflection points are where the second deritivative is zero.

The rest is algebra.