Using analytical method, find the extreme values of the functions on the interval and where they occur.

1. f(x) = ln(x+1), 0<=x<=3

2. f(x) = sin(x+ pi/4), 0<=x<=(7pi)/4

Thanks in advance.

1. d/dx ln (x+1)+d/dx(x+1)

*remember to use the chain rule
=1/x+1 + 1

endpoints: plug in the numbers of the interval
f(0)=ln1=0
f(3)=ln(3+1)=ln4=1.386

maximum:ln4 at x=3
minimum:0 at x=0

To find the extreme values of a function using the analytical method, you need to find the critical points and endpoints of the given interval. Once you have identified these points, you can evaluate the function at these points to determine the extreme values and where they occur.

Let's go through each function step by step:

1. f(x) = ln(x+1), 0 <= x <= 3:
To find the critical points, we need to differentiate the function with respect to x. So, calculate f'(x):

f'(x) = (1 / (x+1))

Setting f'(x) = 0, we obtain:

1 / (x+1) = 0

There is no real number solution for this equation. Therefore, there are no critical points within the interval (0,3).

Next, we evaluate the function at the endpoints of the interval (0 and 3) to determine the extreme values:

f(0) = ln(0+1) = ln(1) = 0
f(3) = ln(3+1) = ln(4)

So, the extreme values for the function f(x) = ln(x+1), on the interval 0 <= x <= 3, are 0 and ln(4). The extreme values occur at x = 0 and x = 3.

2. f(x) = sin(x + pi/4), 0 <= x <= (7pi)/4:
To find the critical points, we differentiate the function with respect to x. So, calculate f'(x) as follows:

f'(x) = cos(x + pi/4)

To find the critical points, set f'(x) = 0:

cos(x + pi/4) = 0

Solving this equation, we get:

x + pi/4 = pi/2 or 3pi/2

x = pi/2 - pi/4 or 3pi/2 - pi/4

x = pi/4 or 5pi/4

These are the critical points within the given interval.

Next, evaluate the function at these critical points and the endpoints to determine the extreme values:

f(pi/4) = sin(pi/4 + pi/4) = sin(pi/2) = 1
f(5pi/4) = sin(5pi/4 + pi/4) = sin(3pi/2) = -1
f(0) = sin(0 + pi/4) = sin(pi/4) = sqrt(2)/2
f((7pi)/4) = sin((7pi)/4 + pi/4) = sin(pi) = 0

The extreme values for the function f(x) = sin(x + pi/4), on the interval 0 <= x <= (7pi)/4, are 1, -1, sqrt(2)/2, and 0. These extreme values occur at x = pi/4, x = 5pi/4, x = 0, and x = (7pi)/4.