# How do you find the maximum value of #f(x)=2sin(x)+cos(x)#?

Range is

f'=2 cos x - sin x = 0, when 2 cos x = sin x that gives

x = arc tan 2. The principal value is in Q1. Indeed, there are general

values in Q1 and Q3.

for Q3 values, as both sin x and cos x are negative in Q3.

The maximum is obtained when tan x = 2, with x in Q1. And this is

2sin x + cos x , with tan x = 2

Alternative method sans differentiation:

By signing up, you agree to our Terms of Service and Privacy Policy

To find the maximum value of ( f(x) = 2\sin(x) + \cos(x) ), you can use calculus.

- Take the derivative of ( f(x) ) with respect to ( x ) to find critical points.
- Set the derivative equal to zero and solve for ( x ).
- Once you have the critical points, evaluate ( f(x) ) at these points as well as at the endpoints of the domain to find the maximum value.

Here's the breakdown:

- ( f'(x) = 2\cos(x) - \sin(x) )
- Set ( f'(x) = 0 ): ( 2\cos(x) - \sin(x) = 0 )
- Solve for ( x ): ( \sin(x) = 2\cos(x) ) ( \tan(x) = 2 ) ( x \approx 1.107 ) (using inverse tangent function)
- Evaluate ( f(x) ) at the critical point and endpoints: ( f(1.107) ) and ( f(0) ) and ( f(2\pi) )
- Compare the values obtained in step 4 to find the maximum value of ( f(x) ).

By signing up, you agree to our Terms of Service and Privacy Policy

- In the first Mean Value Theorem #f(b)=f(a)+(b-a)f'(c), a<c<b, f(x) =log_2 x, a=1 and f'(c)=1. How do you find b and c?
- Find critical numbers for f(x)= x(x-2)^(-3) .explain why x= 2 is not one?
- How do you find the maximum value of #y = −2x^2 − 3x + 2#?
- Is #f(x)= x/sinx # increasing or decreasing at #x=-pi/6 #?
- How do you find the critical points for the inequality #(2x+1)/(x-9)>=0#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7