Determine the open intervals on which the graph of f(x) = -7x^2 + 8x + 1 is concave downward or concave upward.
The second derivative is just -14, so I don't know what to do with that.
Ok, so the second derivative is negative and you know that if the 2nd derivative is negative the function is concave downwards.
Of course you should recognize that we are dealing with a quadratic function, a parabola, which opens downwards for all values of x
To determine the concavity of the graph of a function, we need to analyze the sign of its second derivative. In this case, you correctly found that the second derivative of f(x) = -7x^2 + 8x + 1 is -14.
The sign of the second derivative tells us whether the graph is concave upward, concave downward, or neither.
- A positive second derivative (+) indicates a concave upward graph.
- A negative second derivative (-) indicates a concave downward graph.
Since the second derivative, in this case, is -14 (negative), the graph of f(x) = -7x^2 + 8x + 1 is concave downward for all values of x.
Therefore, there are no open intervals where the graph is concave upward. It is concave downward over the entire domain.