Find the absolute maximum and absolute minimum values of f on the given interval.
f(x) = (e^−x)*(− e^−6x), [0, 1]
well, (e^−x)*(−e^−6x) = -e^-7x
as you know exponentials have no local max/min, so the extrema are at the ends of the interval.
That should pose no difficulty, if you sketch the graph.
That's not the derivative of the function tho Steve.
well, the derivative is 7e^-7x
where is that zero?
no the derivative is not 7e^-7x
when the function is e^-x-e^-6x
To find the absolute maximum and absolute minimum values of a function f(x) on a given interval, we can follow these steps:
Step 1: Find the critical points of f(x) strictly in the given interval [0, 1]. These are the values of x where the derivative of f(x) is either zero or undefined.
Step 2: Evaluate the function at the critical points obtained in step 1 and at the endpoints of the interval [0, 1].
Step 3: Compare the values obtained in step 2 to determine the absolute maximum and absolute minimum.
Now let's apply this process to the given function f(x) = e^-x * - e^-6x on the interval [0, 1]:
Step 1: Find the derivative of f(x) with respect to x:
f'(x) = (-e^-x) * - e^-6x * (1) + e^-x * -6 * -e^-6x
= e^-x * e^-6x + 6e^-x * e^-6x
= e^(-x-6x) + 6e^-7x
= e^-7x + 6e^-7x
= 7e^-7x
To find the critical points, we need to solve the equation f'(x) = 0:
7e^-7x = 0
Since e^-7x is always positive and never zero, the derivative f'(x) is never zero on the interval [0, 1]. Therefore, there are no critical points within this interval.
Step 2: Evaluate the function at the endpoints and the critical points:
f(0) = e^0 * - e^0 = -1
f(1) = e^-1 * - e^-6 = -e^-7
Step 3: Compare the values obtained:
We have f(0) = -1 and f(1) = -e^-7.
Since e^-7 is a positive value between 0 and 1, we can conclude that f(1) = -e^-7 is the maximum value of the function on the interval [0, 1]. The absolute maximum value is -e^-7.
Similarly, since f(0) = -1 is the lowest value of the function on the interval [0, 1], it is also the absolute minimum value.
Therefore, the absolute maximum value is -e^-7 and the absolute minimum value is -1 for the function f(x) = e^-x * - e^-6x on the interval [0, 1].