Let f(x) = l (sinx) - 1/2 l. <--- absolute value of sinx - 1/2
The maximum value attained by f is.
Range of sin ( x ) = [ -1 , 1 ]
When sin ( x ) = - 1 then:
abs ( sin ( x ) - 1 / 2 ) = abs ( - 1 - 1 / 2 ) = abs ( - 1.5) = + OR - 1.5
When sin ( x ) = 1 then:
abs ( sin ( x ) - 1 / 2 ) = abs ( 1 - 1 / 2 ) = abs ( 0.5) = + OR - 0.5
The maximum value = 1.5
Correction:
abs ( - 1.5) = 1.5
abs ( 0.5) = 0.5
Well, let's take a closer look at the function f(x) = |sinx - 1/2|. To find the maximum value of f, we need to find the highest point on the graph.
Now, the absolute value function |x| just gives you the distance of x from zero. So in this case, |sinx - 1/2| represents the vertical distance between sinx and 1/2.
As we know, sinx oscillates between -1 and 1, and 1/2 is right in the middle. So the largest possible distance between sinx and 1/2 would occur when sinx reaches either its maximum of 1 or its minimum of -1.
When sinx = 1, the absolute value becomes |1 - 1/2| = |1/2| = 1/2. And when sinx = -1, the absolute value becomes |-1 - 1/2| = |-3/2| = 3/2.
Therefore, the maximum value attained by f(x) is 3/2. So let's give it a round of applause for reaching the top!
To find the maximum value attained by f(x), we need to determine the critical points of the function.
The absolute value function |x| is equal to x when x is non-negative, and -x when x is negative. In this case, since we have |sinx - 1/2|, we need to consider two cases:
1. sinx - 1/2 ≥ 0:
In this case, the absolute value function becomes sinx - 1/2. To find the maximum value, we need to find the maximum value of sinx. The maximum value of sinx is 1, so the maximum of sinx - 1/2 when sinx - 1/2 ≥ 0 is 1 - 1/2 = 1/2.
2. sinx - 1/2 < 0:
In this case, the absolute value function becomes -(sinx - 1/2) = 1/2 - sinx. Again, we need to find the maximum value of 1/2 - sinx, which occurs when sinx is minimized. The minimum value of sinx is -1, so the maximum of 1/2 - sinx when sinx - 1/2 < 0 is 1/2 - (-1) = 3/2.
Now, we compare the maximum values obtained in the two cases:
- The maximum value when sinx - 1/2 ≥ 0 is 1/2.
- The maximum value when sinx - 1/2 < 0 is 3/2.
Since 3/2 is greater than 1/2, the maximum value attained by f(x) is 3/2.
To find the maximum value attained by f(x), we need to find the critical points of the function and then check whether they correspond to a maximum.
First, let's find the critical points of f(x) by taking its derivative. Using the chain rule, the derivative of f(x) is:
f'(x) = l cosx l
A critical point occurs when f'(x) = 0. Since the absolute value of a value cannot be negative, we can ignore the negative sign. So, we have:
cosx = 0
The solutions to this equation occur when x is equal to π/2 plus any multiple of π (i.e., π/2, 3π/2, 5π/2, etc.).
Next, we need to check whether these critical points correspond to a maximum. To do this, we can analyze the behavior of f(x) around these points.
Consider a point x = π/2. As x approaches π/2 from the left (i.e., x < π/2), the function f(x) approaches -1/2. As x approaches π/2 from the right (i.e., x > π/2), the function f(x) approaches 1/2. Therefore, the function has a maximum value of 1/2 at x = π/2.
Similarly, for x = 3π/2, 5π/2, and so on, the function also has a maximum value of 1/2.
Hence, the maximum value attained by f(x) is 1/2.