In the App[br] Adjust the values of the coefficients a, b, and c via the sliders or input boxes.[br][br]Notice that when the graph of [i]f[/i]([i]x[/i]) is increasing its derivative will be positive (above the [i]x[/i]-axis), and when the graph of [i]f[/i]([i]x[/i]) is decreasing its derivative will be negative (below the[i] x[/i]-axis). When the graph of [i]f[/i]([i]x[/i]) is at the extremum (maximum if [i]a[/i] < 0 and minimum if [i]a[/i] > 0), then the derivative will be 0 (on the [i]x[/i]-axis). [br][br]Show the graph and formula for the derivative by checking its checkbox. Can you see a general pattern for the formula of the derivative? Test out your conjecture by trying different values of [i]a, b[/i], and [i]c[/i]. [br][br]The graph and formula for f "(x) can also be displayed. Note that if a > 0 then the graph of the second derivative is always above the x-axis (positive valued) and the graph of the first derivative is increasing and the graph of the original function is concave up. If a < 0 then the second derivative is negative, the first derivative is decreasing, and the original function is concave down.[br][br]Check Derivative Rule to see the shortcut rule for derivatives of quadratic functions. Check Proof to see an algebraic proof of this derivative rule.[br]