The function f has derivatives of all orders for all real numbers x. Assume f(2)=-3, f'(2)=5, f''(2)=3, and f'''(2)=-8.
The fourth derivative of f satisfies the inequality, the (absolute value of f''''(x)) <=3 for all x in the closed interval [1.5, 2]. Use the Lagrange error bound on the approximation to f(1.5) found in part (a) to explain why f(1.5) does not equal -5.
The part (a) was where I wrote the third-degree Taylor polynomial for f about x=2 and approximate f(1.5).
I had -3+5(x-2) +3(x-2)^2/2-8(x-2)^3/6, which was -4.958333 when f(1.5).
Please help me answer this question.
To explain why f(1.5) does not equal -5 using the Lagrange error bound on the approximation of f(1.5), we need to consider the range of possible error in the approximation. The Lagrange error bound provides an upper bound for the absolute value of the error between the approximation and the actual value of the function.
In this case, you have found the third-degree Taylor polynomial for f about x=2 and approximated f(1.5) using this polynomial. The value you obtained, -4.958333, is an approximation to f(1.5) using the polynomial.
Now, we can use the Lagrange error bound to determine the maximum possible error in this approximation. The Lagrange error bound for a Taylor polynomial of degree n is given by:
|f(x) - Pn(x)| <= (M * |x - a|^(n+1))/(n+1)
where f(x) is the actual function, Pn(x) is the nth-degree Taylor polynomial, M is the maximum value of the (n+1)th derivative on the interval [a, x], and |x - a| is the distance between the point of approximation (x) and the center of the polynomial (a).
In this case, since the fourth derivative of f satisfies the inequality |f''''(x)| <= 3 for all x in the closed interval [1.5, 2], we know that the maximum value of the fourth derivative on this interval is 3 (as it is an upper bound).
Since the Lagrange error bound provides an upper bound for the error, we can use it to determine an upper bound for the difference between the actual value of f(1.5) and the approximation -4.958333.
Plugging the values into the Lagrange error bound formula, we have:
|f(1.5) - (-4.958333)| <= (3 * |1.5 - 2|^4)/(4+1)
Simplifying, we get:
|f(1.5) + 4.958333| <= (3 * 0.5^4)/5
Now, the left side of this inequality represents the possible range of values for f(1.5), whereas the right side represents the upper bound given by the Lagrange error bound.
Since we want to determine if f(1.5) could be equal to -5, we can substitute -5 into the inequality:
|-5 + 4.958333| <= (3 * 0.5^4)/5
Simplifying, we have:
|-0.041667| <= 0.003
Since the absolute value of -0.041667 is less than 0.003, we can conclude that f(1.5) does not equal -5 based on the Lagrange error bound. The upper bound given by the Lagrange error bound is significantly larger than the difference between -0.041667 and 0.003, indicating that f(1.5) falls within a range of values that is not equal to -5.