Yes it's correct.
find all values of the constants a and b for which the function
f(x)=ax, x<2 and ax^2-bx+3, x greater than or equal to 2
will be differentiable for all values of x
so what i first did was to find the values of a and b for which the function is continuous, so i took the left sided and right sided limits as they approached 2
for the left sided limit i got 2a, and for the right sided limit i got 4a-2b+3
i set them equal to each other 2a=4a-2b+3 and rearranged to get -2a+2b=3
next i found the left sided and right sided limits of the derivative of the function as it approached 2
for the left side limit i got 2, and for the right side limit i got 4a-b
setting them equal to each other to get 4a-b=2, i then have a system of equations to solve
solving i get a=7/6 and b=8/3
i just want someone to look over my work and verify that its right or wrong, or if this is how someone would go about solving a problem like this, i wasnt really sure, my first guess was that i had to use the limit definition of the derivative to somehow get the answer, but i wasnt really sure if that would work so i tried the method used above.
Don't forget to study if f is differentiable at x_0 =/= 2 as the question says.
For example if x_0 < 2 there exists a neighborhood V of x_0 such that f : V -> IR , f ( x ) = a x so, f is an elemental function on V and by a well known theorem f' ( x ) = a etc.