f(x) = 7 x^2 - 4 x.

Find all extreme values (if any) of f and determine the values of x at which these values occur.

Question

f(x) = 7 x^2 - 4 x.

Find all extreme values (if any) of f and determine the values of x at which these values occur.

anonymous · Answer

1) Find the derivative
$$f(x) =  7x ^{2}-4x$$
$$f'(x) = 14x - 4$$

2) Set the derivative = 0 to find the critical points
$$14x-4 = 0$$
$$x = 2/7$$

3) Set up your intervals to determine increasing/decreasing
$$(-\infty, 2/7) and (2/7, \infty)$$

3a) Find a test point in each interval and substitute into your first derivative to determine if increasing or decreasing. 
If you get from increasing to decreasing, you have a max. 
If you get from decreasing to increasing, you have a min.

In the first interval, you get a negative (which means the graph is decreasing from -infinity to 2/7). 
   For the second interval, you get a positive (which means the graph is increasing from 2/7 to infinity).

Therefore, this means you have a minimum at x = 2/7.

If you'd like to know the y-coordinate, simply plug in 2/7 in for x into your original function.

Hope this helps! Good luck!