00:01
In this problem, we are provided with the function f of x which equals to 10 minus 5 times e raised to the part negative x.
00:09
In subpart a, we are asked to find out the critical numbers of this function.
00:15
So for the critical numbers, we first require the first derivative of the function.
00:19
The derivative of any constant is 0 and the derivative of e raised to the part negative x is e raised to the part negative x times negative 1.
00:27
So this simplifies to 5 times e raised to the part negative x.
00:31
Next we equate this to 0 and we solve for x.
00:35
So we have 5 times e raised to the power negative x to be equal to 0.
00:39
But we know that e raised to the power negative x is never equal to 0.
00:45
So here it implies that there is no such value of x such that the value of f prime of x equals to 0.
01:00
So therefore none of the points are critical points for the given function.
01:06
So this is the required answer for subpart a.
01:10
Next, in subpart b, we are asked to find out the interval where the function is increasing.
01:19
So here, since we have f prime of x to be equal to 5 times e raised to the par negative x, which is always greater than 0, it implies that the function is increasing in the entire domain negative infinity to positive infinity.
01:34
So this is the required answer for subpart b.
01:39
Next, in subpart c, we are asked to find out the x coordinate of the local minima of the function.
01:51
So the local minima of the function is the point x where the function, f of x, changes from being decreasing to increasing.
02:05
But since the function is always increasing, there is no such local minima for the provided function f of x.
02:15
So this is the required answer for subart c...