Find the Maclaurin series for f(x) = cos(x²) and use it to determine f⁽⁴⁾(0).

Maclaurin series is nothing but Taylor series about the point x = 0.

Answer: cos (x²) = 1 + x⁴/2! + x⁸/4! + x¹²/6! + x¹⁶/8! + ..., the value of f⁽⁴⁾(0) = 12

Let us write the Maclaurin series for f(x) = cos(x²) and use it to determine f⁽⁴⁾(0).

Explanation:

Maclaurin series of the function f(x) is given by

f(x) = f(0) + f'(0)x + f"(0) x² / 2! + f'"(0) x³ / 3! + ... + f⁽ⁿ⁾(0) xⁿ / n! + ... ------- (1)

Using the definition of Maclaurin series, we can write cos x as

cos x = 1 + x²/2! + x⁴/4! + x⁶/6! + x⁸/8! + ...

Replace x by x², we get 

cos (x²) = 1 + x⁴/2! + x⁸/4! + x¹²/6! + x¹⁶/8! + ...-------- (2)

f⁽⁴⁾(0) is the 4^th derivative of f(x) evaluated at x = 0.

Comparing the coefficients of x⁴ from equations (1) and (2), we get

f⁽⁴⁾(0) / 4! = 1/2!

f⁽⁴⁾(0) = 4! / 2! = 12

So, f⁽⁴⁾(0) = 12