Find the linearization l(x) of the function at a. f(x) = x⁴ + 6x², a= -1.

Solution:

Linearization is an effective method for approximating the output of a function at any point based on the value and slope of the function, given that is differentiable on (or ) and that is close to.

In short, linearization approximates the output of a function near.

The linearization of f(x) = x⁴ + 6x² at a = -1 is one form of the equation of the line tangent to the graph of f at the point (-1,f(-1))

The linearization of the function is given by

Y = f(a) + (x-a) f^-1(a)

Where f^-1(x) is the derivative of f(x)

I.e f^-1(x) = 4x³ + 6(2x) {since the derivative of xⁿ is nx^n-1}

⇒ f^-1(x) = 4x³ + 12x

∴ f^-1(a) = f^-1(-1) = 4(-1)³+ 12(-1) = -4 + 12(-1)

∴ f^-1(a) = f^-1(-1) = -4 - 12

f^-1(a) = f^-1(-1) = -16

f(x) = x⁴+6x² ⇒ f(a) = f(-1) = (-1)⁴+6(-1)² = 1 + 6(1) {since (-1)even= +1}

f(-1) = 1 + 6 = 7

∴The linearization of the function is given by

Y = 7 + [x - (-1) ](-16)

I.e Y = 7 + [x +1](-16)

on simplifying

Y = 7 - 16x - 16

Y = -16x - 9

---

Find the linearization l(x) of the function at a. f(x) = x⁴ + 6x², a=-1.

Summary:

The linearization of the function f(x) at a. f(x) = x⁴ + 6x², a= -1 is Y = -16x - 9