Answer:
The absolute minimum of the function [tex]f(x) = x^{4} - 4\cdot x - 3[/tex] occurs at [tex]x = 1[/tex] and is [tex]f(1) = -6[/tex].
Explanation:
Statement is incorrect. Correct statement is presented below:
Given the function [tex]f(x) = x^{4}-4\cdot x - 3[/tex], determine the absolute minimum value of [tex]f[/tex] on the closed interval [tex](-2, 4)[/tex]. First, we determine the first and second derivatives of the function.
First Derivative
[tex]f'(x) = 4\cdot x^{3}-4[/tex] (1)
Second Derivative
[tex]f''(x) = 12\cdot x^{2}[/tex] (2)
By equalizing (1) to zero, we solve for [tex]x[/tex]:
[tex]4\cdot x^{3}-4 = 0[/tex]
[tex]x^{3}= 1[/tex]
[tex]x = 1[/tex]
And we evaluated this result in (2):
[tex]f''(1) = 12[/tex]
According to criteria of the Second Derivative Test, we conclude that value of [tex]x[/tex] leads to an absolute minimum. The value of the absolute minimum is:
[tex]f(1) = -6[/tex]
The absolute minimum of the function [tex]f(x) = x^{4} - 4\cdot x - 3[/tex] occurs at [tex]x = 1[/tex] and is [tex]f(1) = -6[/tex].
