Final answer:
The linear approximation of the function f(x)=x⁸+5x² at a=-1 is -6 - 2(x+1).
Step-by-step explanation:
The linear approximation of the function f(x)=x⁸+5x² at a=-1 can be found using the formula for linear approximation:
f(x) ≈ f(a) + f'(a)(x-a)
First, we need to find the derivative of the function:
f'(x) = 8x⁷ + 10x
Next, evaluate the function and its derivative at x = -1:
f(-1) = (-1)⁸ + 5(-1)² = -6
f'(-1) = 8(-1)⁷ + 10(-1) = -2
Finally, substitute these values into the linear approximation formula:
f(x) ≈ -6 - 2(x+1)
Thus, the linear approximation of the function f(x)=x⁸+5x² at a=-1 is -6 - 2(x+1).