Find the intervals on which f is increasing and the intervals on which it is decreasing.

Question

f(x)= 1-x + 3x^2

Answer 1

y' = 6x-1 = 0

x = 1/6

So x<1/6 the function decreases and for x>1/6 the function increases

Answer 2

Remember that the original function f(x) is increasing wherever the derivative f'(x) is positive, and f(x) decreases wherever f'(x) is negative.

We first take the derivative with the power rule: f'(x) = -1+6x. Graphically, this function is a line with a positive slope. The x-intercept is at x = 1/6. Since the slope is positive, f'(x) will be positive to the right of x = 1/6, and negative to the left of x = 1/6.

Putting this all together, the original function will be increasing on (1/6, infty), and decreasing on (-infty, 1/6). In terms of the original function, x=1/6 will correspond to the vertex (minimum) of the parabola.