The function defined by f(x) = 4⁴ – 2x + 1 is increasing for

The correct option is x > 1/2.

To understand why this is correct, we analyze the given function. The expression in the question, written as $f\left(x\right)=4x^2-4x+1$ (which can also be written in its factored form as the perfect square $(2x-1{)}^2$), represents a quadratic function.

A function is increasing on an interval where its first derivative is positive, i.e., where:
$f^{\prime }\left(x\right)>0$

First, we find the derivative of $f\left(x\right)$ with respect to $x$ using the power rule:
$f^{\prime }\left(x\right)=\frac{d}{dx}(4x^2-4x+1)$
$f^{\prime }\left(x\right)=8x-4$

Next, to determine the interval where the function is increasing, we set the first derivative to be greater than zero:
$8x-4>0$

Solving this inequality for $x$:
$8x>4$
$x>\frac{4}{8}$
$x>\frac{1}{2}$

Thus, the function is strictly increasing for the interval x > 1/2.