Question #311455

The function 𝑓(𝑥) is defined as 𝑓(𝑥)=𝑎+𝑏cos𝑥, where 𝑎 and 𝑏 are constants. The range of 𝑓(𝑥) is given by −6≤𝑓(𝑥)≤2.

a. Find the values of 𝑎 and 𝑏

b. Solve the equation 𝑓(𝑥)=0 for 0°≤𝑥≤360°

c. Sketch the graph of 𝑦=𝑓(𝑥) for 0°≤𝑥≤360°


1
Expert's answer
2022-03-17T05:16:52-0400

a)

Because the min value of the function = -6: a+bcosx=6a+bcosx =-6

max value = 2: a+bcosx=2a+bcosx = 2

Because 1cosx1-1≤cosx≤1 it means min(cosx)=1,max(cosx)=1min(cosx)=-1, max(cosx)=1

Make up a system:{a+b(1)=6a+b1=2\begin{cases} a+b(-1)=-6 \\a+b*1=2 \end{cases}     \implies {ab=6a+b=2\begin{cases} a-b=-6 \\ a+b=2 \end{cases}     \implies {2a=4a+b=2\begin{cases} 2a=-4 \\ a+b = 2 \end{cases}     {a=2b=4\implies\begin{cases} a =-2 \\ b=4 \end{cases}

Answer:f(x)=2+4cosxf(x)=-2+4cosx


b)

2+4cosx=0,x[0;2π]-2+4cosx=0, x\isin[0;2\pi]

4cosx=2    cosx=24    cosx=124cosx=2\implies cosx=\frac24 \implies cosx = \frac 12


x=+x=\begin{smallmatrix}+\\- \end{smallmatrix} π3+2πn,nZ,Z=\frac\pi3+2\pi n, n\isin Z, Z = set of integer


for n=0:n=0: x=π3x =\frac\pi3 and x=π3x=-\frac\pi3 - not suitable

for n=1:x=π3+2πn =1:x=\frac\pi3+2\pi - not suitable, and x=π3+2π=5π3x=-\frac\pi3 +2\pi = \frac{5\pi}3

for n=1:x=π32πn=-1: x= \frac\pi3-2\pi - not suitable, and x=π32πx=-\frac\pi3-2\pi - not suitable


Answer: x=π3,x=5π3x=\frac\pi3, x=\frac{5\pi}3


c)

The required plot of the graph between the blue and green line in the screenshot


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