Question
Which value of x in the interval π2≤x≤π makes the line tangent to the function f(x)=2–√x+2cosx horizontal
Answers
GPT 3.5
To find the value of x that makes the line tangent to the function f(x) = 2 – √x + 2cosx horizontal, we need to find when the derivative of f(x) is equal to zero.
First, let's find the derivative of f(x):
f'(x) = d/dx (2 – √x + 2cosx)
= -1/(2√x) - 2sinx
We set f'(x) = 0 to find the critical points:
-1/(2√x) - 2sinx = 0
Multiplying through by 2√x gives us:
-1 - 2√xsinx = 0
Rearranging, we have:
2√xsinx = -1
Dividing by 2√x gives us:
sinx/√x = -1/2
To find the value of x in the interval π/2 ≤ x ≤ π that satisfies this equation, we can use a graphing calculator or a table of values. However, in this case, we can recognize that sinx/√x is always positive in this interval since sinx is positive and √x is positive. Therefore, there is no solution in the interval π/2 ≤ x ≤ π.
First, let's find the derivative of f(x):
f'(x) = d/dx (2 – √x + 2cosx)
= -1/(2√x) - 2sinx
We set f'(x) = 0 to find the critical points:
-1/(2√x) - 2sinx = 0
Multiplying through by 2√x gives us:
-1 - 2√xsinx = 0
Rearranging, we have:
2√xsinx = -1
Dividing by 2√x gives us:
sinx/√x = -1/2
To find the value of x in the interval π/2 ≤ x ≤ π that satisfies this equation, we can use a graphing calculator or a table of values. However, in this case, we can recognize that sinx/√x is always positive in this interval since sinx is positive and √x is positive. Therefore, there is no solution in the interval π/2 ≤ x ≤ π.