To determine the values of
a and
b for which the function
f(x)={is differentiable everywhere, we need to ensure two things at the point
x=1 :
Continuity at
x=1.
Equal left-hand and right-hand derivatives at
x=1.
Let's go through these steps:
Continuity at
x=1 :
For
f(x) to be continuous at
x=1, the value of the function from the left-hand side must equal the value from the right-hand side. Thus, we equate:
f(1−)=f(1+)From the left-hand side:
f(1)=12+3(1)+a=1+3+a=4+aFrom the right-hand side:
f(1)=b(1)+2=b+2 Setting these equal gives:
4+a=b+2.Solving for
a :
a=b−2. Differentiability at
x=1 :
The derivative from the left must equal the derivative from the right.
For
x≤1, differentiate
f(x)=x2+3x+a :
f′(x)=2x+3 Evaluating at
x=1 :
f′(1−)=2(1)+3=5.For
x>1, differentiate
f(x)=bx+2 :
f′(x)=b So,
f′(1+)=b.Set the derivatives equal:
5=b.Now substitute
b=5 into the equation we found from continuity:
a=5−2=3. Thus, the function is differentiable everywhere when
a=3 and
b=5.