[amp_mcq option1=”concave upward for all values of x” option2=”concave downward for all x” option3=”concave up only for positive values of x” option4=”concave up for negative values of x” correct=”option1″]
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.