Find the inflection point(s), if any, of the function.
Calculus

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First, we find the second derivative of the function.
g(x) = 4x^4  8x^3 + 2 > g'(x) = (4x^4  8x^3 + 2)' > g'(x) = (4x^4)'  (8x^3)' + (2)'
g'(x) = 4(x^4)'  8(x^3)' + (2)' > g'(x) = 4(4x^41)  8(3x^31) + 0 > g'(x) = 4(4x^3)  8(3x^2)
g'(x) = 16x^3  24x^2
g''(x) = (16x^3  24x^2)' > g''(x) = (16x^3)'  (24x^2)' > g''(x) = 16(3x^31)  24(2x^21)
g''(x) = 48x^2  48x^1 > g''(x) = 48x^2  48x
Then we equal the second derivative to zero and solve for x.
48x^2  48x = 0 > 48x(x  1) = 0 . It is factored, so now we equal each factor to zero and solve for x like this:
48x = 0 > 48x/48 = 0/48 > x = 0
x  1 = 0 > x  1 + 1 = 0 + 1 > x = 1
Now we enter each x value into the original function g(x) in order to find the y coordinate like this.
g(0) = 4(0)^4  8(0)^3 + 2 > g(0) = 0  0 + 2 = 2. The ordered pair (x , y) is: (0 , 2)
g(1) = 4(1)^4  8(1)^3 + 2 > g(1) = 4  8 + 2 =  2. The ordered pair is: (1 , 2)
(0, 2) smaller x
(1, 2) larger x
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