Consider the function f : [�-pi,pi�] --> R given by f(x) = e^-x*sin(x).
(a) Calculate local maxima/minima and points of in
ection of f(x).
Let's see what this looks like
https://www.wolframalpha.com/input/?i=plot+y+%3D++%28e%5E-x%29%28sin%28x%29%29
f ' (x) = e^-x(cosx) - sinx (e^-x)
= 0 for max/min
e^-x(cosx - sinx)=0
e^-x = 0 ---> not possible
or
cosx = sinx
tanx = 1
x = pi/4 or -3pi/4
sub that back in original to find the corresponding max/min
How do you get the point of inflections?
To calculate the local maxima/minima and points of inflection of a function, you need to find its critical points and determine the behavior of the function around those points.
First, we need to find the critical points of the function. Critical points occur where the derivative of the function is either zero or undefined. Let's start by finding the derivative of f(x).
f(x) = e^(-x) * sin(x)
Using the product rule, we can differentiate f(x) as follows:
f'(x) = (e^(-x))' * sin(x) + e^(-x) * (sin(x))'
Taking the derivatives of e^(-x) and sin(x):
f'(x) = -e^(-x) * sin(x) + e^(-x) * cos(x)
Now set f'(x) equal to zero and solve for x:
0 = -e^(-x) * sin(x) + e^(-x) * cos(x)
Divide through by e^(-x):
0 = -sin(x) + cos(x)
Rearranging the equation:
sin(x) = cos(x)
Dividing through by cos(x):
tan(x) = 1
This equation is satisfied when x = pi/4 or x = 5pi/4.
Next, we need to check for points of inflection by finding the second derivative of f(x). Differentiating f'(x) from above:
f''(x) = -(-sin(x) + cos(x)) + sin(x) + cos(x)
Simplifying:
f''(x) = 2cos(x)
Setting f''(x) equal to zero, we have:
0 = 2cos(x)
This equation is satisfied when x = pi/2 or x = 3pi/2.
Now we can summarize the critical points and points of inflection we found:
Critical points:
- x = pi/4
- x = 5pi/4
Points of inflection:
- x = pi/2
- x = 3pi/2
To determine whether these critical points or points of inflection correspond to local maxima, minima, or points of inflection, we need to analyze the behavior of the function in their vicinity. You can do this by looking at the first and second derivative tests or by graphing the function.