The answer is -1/2 but how?
Which of the following is the minimum value of the function f(x)= sinxcosx?
a) 0
b) -1/2
c) -1
d) does not exist
f(x) = sinxcosx
= (1/2) sin (2x) by a common identity
At this point I have my answer.
y = (1/2)sin(2x) has an amplitude of 1/2
that is , its maximum is 1/2 and its minimum is -1/2
thanks!
To find the minimum value of the function f(x) = sin(x)cos(x), we can use calculus. The minimum or maximum of a function usually occurs at critical points, which are the points where the derivative of the function is either zero or undefined.
First, take the derivative of f(x) with respect to x. The derivative of sin(x)cos(x) can be found using the product rule of differentiation.
Let's denote sin(x) as u and cos(x) as v:
u = sin(x)
v = cos(x)
Now, applying the product rule:
f'(x) = u'v + uv'
= (cos(x))(cos(x)) + (sin(x))(-sin(x))
= cos^2(x) - sin^2(x)
To find the critical points of f(x), we need to set f'(x) equal to zero:
cos^2(x) - sin^2(x) = 0
Applying the trigonometric identity cos^2(x) - sin^2(x) = cos(2x), we have:
cos(2x) = 0
The critical points occur when cos(2x) equals zero. This means that 2x is equal to π/2 or 3π/2 plus any multiple of π. Simplifying further:
2x = π/2 + nπ/2, where n is an integer
Dividing both sides by 2, we get:
x = π/4 + nπ/4, where n is an integer
Now we need to evaluate f(x) at these critical points to find the minimum value. Substitute each critical point into the original function f(x) = sin(x)cos(x):
f(π/4 + nπ/4) = sin(π/4 + nπ/4)cos(π/4 + nπ/4)
Simplifying the trigonometric functions using the sum formula for sin(x + y) and cos(x + y), we have:
f(π/4 + nπ/4) = (1/√2)(1/√2) = 1/2
Therefore, the minimum value of the function f(x) = sin(x)cos(x) is 1/2.
Therefore, the correct answer is none of the given options (a), (b), (c), or (d), as the minimum value is not -1/2 but 1/2.