ind the linearization L(x) of the function at a = -1.
f(x) = x^4 + 6x^2
To find the linearization of a function at a given point, we need to know the value of the function at that point, as well as its derivative.
The value of the function at the point a = -1 can be calculated by substituting -1 into the function:
f(-1) = (-1)^4 + 6(-1)^2
= 1 + 6
= 7
Now, let's find the derivative of the function. The derivative of a function is calculated using the power rule. For our function f(x) = x^4 + 6x^2, we can differentiate each term separately:
f'(x) = d/dx (x^4) + d/dx (6x^2)
= 4x^3 + 2(6x)
= 4x^3 + 12x^2
Next, we can use the point (a, f(a)) and the derivative f'(x) to construct the equation for the linearization L(x):
L(x) = f(a) + f'(a)(x - a)
Substituting the values we have:
L(x) = f(-1) + f'(-1)(x - (-1))
= 7 + (4(-1)^3 + 12(-1)^2)(x + 1)
= 7 + (4(-1) + 12)(x + 1)
= 7 + (-4 + 12)(x + 1)
= 7 + 8(x + 1)
= 7 + 8x + 8
= 8x + 15
Therefore, the linearization of the function f(x) = x^4 + 6x^2 at a = -1 is L(x) = 8x + 15.