Question

How do you find the values of c and d that make the following function continuous for all x given `f(x) = 9x` if x<1, `f(x) = cx^2+d` if `1<=x<2` and `f(x) = 3x` if `x>=2`?

Answer 1

`c=-1, and, d=10`.

Explanation:

Let us name the Intervals

`x<1" as "I_1, 1lexlt2" as "I_2," and, "xge2" as "I_3`.

On these Intervals `f` is defined as polynomials, which, we know,

are continuous on these intervals.

So, if `f` has to be made continuous over the whole of `RR`, it

has to be continuous at the joining points of these intervals; i.e., to

say, it must be continuous at `x=1, and, x=2`.

Now, for continuity at `x=1`, we must have,

`lim_(xrarr1-) f(x) = f(1) = lim_(xrarr1+) f(x)...............(1)`.

`"As "xrarr1-, x<1rArrf(x)=9xrarr 9.`

`:.lim_(xrarr1-) f(x)=9...........................................(1.i)`

On the similar lines, we have,

`lim_(xrarr1+) f(x)=c+d=f(1)............................(1.ii)`

`:." From "(1), (1.i) and, (1.ii), c+d=9.....................(I)`

Considering the continuity of `f" at "x=2`, we get,

`4c+d=6.........................................................................(II)`

Solving `(I)" and "(II)," we get, "c=-1, and, d=10`.

Enjoy Maths.!