If a,b,c,d, and x represent positive real numbers, what is the maximum value of the function f(x)= a[cosb(x+c)]-d?
the cos(anything) has value between -1 and +1
so the max value of cosb(x+c) = 1,
the b controls the frequency, and +c moves it to the left c units. This does not affect the maximum
then the max value of a[cosb(x+d)] is 'a'
but then the function is moved down d units
so the max value of your function is a-d
thank you! that really helped
To find the maximum value of the function f(x) = a[cos(bx + c)] - d, we need to determine the critical points and identify their nature. Here's how:
Step 1: Find the derivative of f(x) with respect to x.
f'(x) = -ab*sin(bx + c)
Step 2: Set the derivative equal to zero and solve for x to find the critical points.
- ab*sin(bx + c) = 0
sin(bx + c) = 0
Step 3: Solve for x using the equation sin(bx + c) = 0.
Since sin(0) = 0, we have two cases:
Case 1: bx + c = 0
x = -c/b
Case 2: bx + c = π
x = (π - c)/b
Step 4: Find the second derivative of f(x):
f''(x) = -ab^2*cos(bx + c)
Step 5: Substitute the critical points (x = -c/b and x = (π - c)/b) into the second derivative to identify their nature.
a) When x = -c/b:
f''(-c/b) = -ab^2*cos(b*(-c/b) + c) = -ab^2*cos(0) = -ab^2
Since this value is negative, the critical point x = -c/b is a local maximum.
b) When x = (π - c)/b:
f''((π - c)/b) = -ab^2*cos(b*((π - c)/b) + c) = -ab^2*cos(π) = ab^2
Since this value is positive, the critical point x = (π - c)/b is a local minimum.
Step 6: Determine the maximum value.
Since f(x) has only one critical point that is a local maximum, the maximum value occurs at x = -c/b.
Thus, the maximum value of the function f(x) = a[cos(bx + c)] - d is obtained when x = -c/b.
To find the maximum value of the function f(x) = a[cosb(x+c)] - d, you can follow the steps below:
Step 1: Understand the function
In this case, you have a trigonometric function that involves parameters a, b, c, and d. The maximum value of the function will occur when the cosine function is at its maximum value of 1.
Step 2: Determine the value of x that maximizes the cosine function
Since the cosine function completes one full cycle (period) in 2π radians or 360 degrees, we need to find the value of x that makes the expression inside the cosine function equal to 0 or a multiple of 2π.
Setting b(x + c) equal to 0:
b(x + c) = 0
x + c = 0 (Dividing both sides by b)
x = -c
So, the maximum value of the cosine function occurs at x = -c.
Step 3: Substitute x = -c into the function f(x)
Substituting x = -c into the original function f(x):
f(-c) = a[cosb(-c + c)] - d
f(-c) = a[cosb(0)] - d
f(-c) = a(cos(0)) - d
f(-c) = a(1) - d
f(-c) = a - d
Therefore, the maximum value of the function f(x) = a[cosb(x+c)] - d is equal to a - d.