Question

for what value of a and b will f(x)={ax,x<2 ax^2-bx+3,x>=2 be differentiable for all values of x. plz help with solving this.

Answer 1

To do all these type of questions the function has to be differentiable at the point where definition changes as it is differentiable at all other points.So at x=2 LHD=RHD
LHD=a RHD=2ax-b=4a-b so here one relation a=4a-b i.e, 3a=b;
Now obviously if it is differentiable it must be continuous. So LHL=RHL
2a=4a-2b+3 so 2a+3=2b. Solve these to find a and b.