To find the value of [tex]\( x \)[/tex] at the points of inflection for the curve defined by the function [tex]\( f(x) = 3x^4 - 4x^3 + 1 \)[/tex], let's follow these detailed steps:
1. First Derivative:
First, we need to find the first derivative of the function, [tex]\( f'(x) \)[/tex]:
[tex]\[
f(x) = 3x^4 - 4x^3 + 1
\][/tex]
Taking the derivative with respect to [tex]\( x \)[/tex]:
[tex]\[
f'(x) = \frac{d}{dx}(3x^4 - 4x^3 + 1) = 12x^3 - 12x^2
\][/tex]
2. Second Derivative:
Next, we find the second derivative [tex]\( f''(x) \)[/tex] by differentiating [tex]\( f'(x) \)[/tex]:
[tex]\[
f'(x) = 12x^3 - 12x^2
\][/tex]
Taking the derivative with respect to [tex]\( x \)[/tex]:
[tex]\[
f''(x) = \frac{d}{dx}(12x^3 - 12x^2) = 36x^2 - 24x
\][/tex]
3. Setting the Second Derivative to Zero:
To find the points of inflection, we set the second derivative equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
36x^2 - 24x = 0
\][/tex]
4. Factoring:
Factor the equation:
[tex]\[
12x(3x - 2) = 0
\][/tex]
5. Solving for [tex]\( x \)[/tex]:
Solve the factored equation for [tex]\( x \)[/tex]:
[tex]\[
12x = 0 \quad \text{or} \quad 3x - 2 = 0
\][/tex]
This gives:
[tex]\[
x = 0 \quad \text{or} \quad x = \frac{2}{3}
\][/tex]
Therefore, the values of [tex]\( x \)[/tex] at the points of inflection for the curve [tex]\( 3x^4 - 4x^3 + 1 \)[/tex] are:
[tex]\[
\boxed{0, \frac{2}{3}}
\][/tex]
The correct answer from the provided options is [tex]\( \boxed{0, \frac{2}{3}} \)[/tex]:
d. [tex]\( 0, \frac{2}{3} \)[/tex]
