Define $F(x)=\begin{cases}\sin x, & -\pi\le x<0,\\ cx+d, & 0\le x<2,\\ \sqrt{5-x}, & 2\le x\le 5,\end{cases}$ …

Mathematics

Problem

Define

F(x)=\begin{cases}\sin x, & -\pi\le x<0,\\ cx+d, & 0\le x<2,\\ \sqrt{5-x}, & 2\le x\le 5,\end{cases}

where

c

and

d

are constants. a. Determine the values of

c

and

d

that make

F

continuous on the closed interval

[-\pi,5]

. b. With the constants found in part (a), justify that there exists at least one value

p

[-\pi,5]

for which

F(p)=1

. Clearly identify an interval on which your argument is valid. c. Prove that, with the same constants, the equation

F(x)=0.5

has at least two distinct solutions in

[-\pi,5]

. Indicate separate intervals containing each solution and justify your claim.