Find the coordinates of the minimum or maximum of the function f(x) = 3x^2 -2x +9 and determine if it's a minimum or maximum.
To find the minimum or maximum we need to find a point on the function where its slope is zero. So, we differentiate f(x) and set it equal to zero: f'(x)=0, f'(x) = 6x -2 = 0, x=1/3 Substitute into f(x) to get the y coordinate: f(1/3) = 3 (1/3)2 - 2(1/3) + 9 = 3/9 - 6/9 +81/9 = 78/9 So we know that at the point (1/3, 78/9), f(x) has a minimum or maximum. But which one is it? For that we look at the second derivate, which will tell us about the curvature of the graph. If the second derivate is positive, the graph is concave up, which implies that the point we just found is a minimum. If the second derivative is negative, the function is concave down, which means that we found the maximum. f''(x) = 6 This is positive so at the point (1/3, 78/9), the function is at a minimum.
