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#?

1 Answer
Jan 9, 2017

Answer:

#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.!