Answer:
A function is concave downward if its second derivative is negative. To find the values of x for which the graph of y = x^4 - 6x^2 + x is concave downward, we need to find the values of x for which the second derivative of y is negative.
The second derivative of y = x^4 - 6x^2 + x is y'' = 12x^2 - 12x
To find the values of x for which y'' is negative, we need to find the roots of the equation 12x^2 - 12x = 0.
We can factor the equation as: 12x(x-1) = 0
which gives us x = 0 and x = 1 as the roots.
Now, we need to test the sign of the second derivative in the intervals between the roots.
In the interval (0,1)
y'' = 12x^2 - 12x is positive, therefore the function is not concave downward in this interval.
In the interval (-infinity,0)
y'' = 12x^2 - 12x is negative, therefore the function is concave downward in this interval.
In the interval (1,infinity)
y'' = 12x^2 - 12x is negative, therefore the function is concave downward in this interval.
Therefore, the values of x for which the graph of y = x^4 - 6x^2 + x is concave downward are all values of x less than 0 or greater than 1.