To determine

To find: The value of m and b.

Answer

The value of m and b are 4 and −4.

Explanation

Given:

The function f(x)={x2         if x≤2 mx+b   if x>2.

Derivative Rule:

(1) Power Rule: ddx(xn)=nxn−1

(2) Sum rule: ddx(f+g)=ddx(f)+ddx(g)

(3) Constant multiple rule: ddx(cf)=cddx(f)

Calculation:

Obtain the derivative of the function f(x) if x<2.

f′(x)=ddx(f(x))=ddx(x2)

Apply the power rule (1) and simplify the terms,

f′(x)=2x2−1=2x

Thus, the derivative of the function f(x) if x<2 is f′(x)=2x.

Obtain the derivative of the function f(x) if x>2.

f′(x)=ddx(f(x))=ddx(mx+b)

Apply the derivative rule (2), (3) and (1),

f′(x)=mddx(x)+ddx(b)=m(1x1−1)+0=m

Thus, the derivative of the function f(x) if x>2 is f′(x)=m.

Obtain the value of m and b.

It is given that the function is differentiable everywhere.

Suppose the function is differentiable at 2. That is, the left hand limit of the function f(x) at 2 is equal to the right hand limit of the function f(x) at 2.

This implies that, f′−(2)=4 and  f′+(2)=m.

Therefore, the value of m=4.

Clearly, the function f(x) is continuous at 2 because every differentiable function is continuous. That is, limx→2f(x)=f(2).

This implies limx→2+f(x)=f(2).

limx→2+f(x)=limx→2+mx+b=m(2)+b=2m+b

Since f(2)=4, 2m+b=4 and substitute 4 for m,

2(4)+b=4b=4−8b=−4

Thus, the value of b=−4.

Therefore, the value of m and b are 4 and −4.