The curve y = x4 is A. concave upward for all values of x B. concave downward for all x C. concave up only for positive values of x D. concave up for negative values of x

The correct answer is A. concave upward for all values of x.

The second derivative of a function tells us whether the function is concave up or concave down. A function is concave up if its second derivative is positive, and concave down if its second derivative is negative.

The second derivative of $y=x^4$ is $y”=4x^3$. Since $4x^3$ is always positive for all real numbers $x$, we can conclude that $y=x^4$ is concave up for all values of $x$.

Here is a graph of $y=x^4$:

[asy]
unitsize(1 cm);

draw((0,0)–(4,16));
draw((0,0)–(0,4));

label(“$x$”, (4,0), E);
label(“$y$”, (0,4), N);

draw(graph(x^4, -2, 2),red);
[/asy]

As you can see, the graph of $y=x^4$ is a parabola that opens upwards. This is a characteristic of a concave up function.