analytic and solid geometry

Question

Given a function f, an interval [a,b] and a value V.  Find a value c in the interval so that f(c)=V.  Apply the Intermediate Value Theorem. 

NOTE: Round-off your answers to 2 decimal places, stating from the least to the highest, if there would be more than one.  For positive answers, do not state the + sign anymore.  For answers less than 1, precede them with 0, example 0.11

Answers:

(a)  c =         ; c = 

(b)  c =   

Answer 1

(a) f(c)=V=4 then c²-c=4. Let c=2, f(2)=4-2=2, f(5)=25-5=20, therefore, since 2<4<20 (4 is between 2 and 20), we know that c exists in the interval.  f(3)=9-3=6, so c is between 2 and 3. f(2.5)=6.25-2.5=3.75 which is less than V. So c is between 2.5 and 3. c²-c-4=0, c=(1±√(1+16))/2. But we need the value between 2.5 and 3, so c=(1+√17)/2=2.56155... f(½(1+√17))=4=V. The other value for c is not in the given interval.

(b) c=π/6, cos(0)=1 and cos(π/2)=0, π/6 is between 0 and π/2.