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

Calculus
Tutor: None Selected Time limit: 1 Day

Jul 15th, 2015

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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^4-1) - 8(3x^3-1) + 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^3-1) - 24(2x^2-1)

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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Jul 15th, 2015

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