Calculus Question

Question

Find the absolute minimum and absolute maximum values of f on the given interval.

f(x) = x − ln 4x, [1/2, 2]

Absolute maximum:

Absolute minimum:

Answer 1

Hi Sonam,

This is the closed interval method. The function f(x) = x - ln(4x) is continuous on [1/2, 2], now we need to find the critical values by taking the derivative and setting it to 0.

f'(x) = 1 - (1/4x)*4 which simplifies to f ' (x) = 1 - (1/x)

Set the derivative to 0 and find x. This will lead you to x=1 (which is your critical value).

Now we use your endpoints and critical value to find the absolute max and absolute min.

f(1/2) = -0.193

f(1) = -0.386

f(2) = -0.079

The smallest value is when x=1 and the largest value is when x=2. The absolute minimum occurs at x=1 and absolute maximum occurs at x = 2.

Answer 2

Here is link to desmos graph referenced in the video. Attach the following to the URL for Desmos:

/calculator/f2c9ar4qt3

UpdateL video is not processing for some reason, here is some text:

f' = 1 - 1/4x (x) = 1 - 1/x

That equals 0 when 1 - 1/x = 0 => x - 1 = 0 => x = 1.

So the x values to test are 1/2 and 2 (endpoints of the closed interval) and 1 (critical number). Check out Desmos to see the y values of the function at those x values.

Answer 3

There's a procedure for finding the absolute min and max of a continuous function on a closed interval. Maybe you've seen it before.

Find the critical points of the function within the interval. Then evaluate the function at each of these critical points and at the endpoints of the interval. The smallest of these values is the min, the largest is the max.