1.8: Complete the square and find an enclosed area

4PM1/1/November/2020 — Question 7 · 13 marks

f(x)=x^2-9x+14.

Given that

\displaystyle f(x)

can be written in the form

\displaystyle (x+a)^2+b

, where

\displaystyle a

and

\displaystyle b

are constants,

(a) find the value of

\displaystyle a

and the value of

\displaystyle b

(2)

(b) Hence, or otherwise, find

(i) the minimum value of

\displaystyle f(x)

(ii) the value of

\displaystyle x

for which this minimum occurs.

(2)

The curve

\displaystyle C

has equation

\displaystyle y=f(x)

. The line

\displaystyle l

has equation

\displaystyle y=x+5

\displaystyle C

and

\displaystyle l

(4)

(d) Use algebraic integration to find the exact area of the finite region bounded by

\displaystyle C

and

\displaystyle l

(5)