#### To determine

**To find:** The linearization L(x) of the function at *a*.

#### Answer

The linearization L(x) of the function at a=4 is L(4)=14x+1.

#### Explanation

**Given:**

The function f(x)=x and a=4.

**Proof:**

The linearization of f(x) at *a* is L(x)=f(a)+f′(a)(x−a) (1)

Substitute a=4 in f(x)=x,

f(4)=4=2

The derivative of the function f(x)=x is computed as follows,

f′(x)=ddx(x)=12x12−1=12x−12=12x

Substitute a=4 in f′(x),

f′(4)=124=12×2=14

Substitute 4 for *a* in the equation (1),

L(x)=f(4)+f′(4)(x−4)

Substitute f(4)=2 and f′(4)=14,

L(x)=2+14(x−4)=2+14x−1=14x+1

Therefore, the linearization L(x) of the function at a=4 is L(4)=14x+1.