Chapter 10.2, Problem 38E

To determine

To justify: The given statement is true or false.

Answer to Problem 38E

The given statement is False.

Explanation of Solution

Given information:

The function is,

  $f(x)=x-\frac{x^3}{3}+\frac{x^5}{5}-\dots \dots .+{(-1)}^n\frac{x^{2n+1}}{2n+1}+\dots \dots$

Calculation:

The given function is, $f(x)=x-\frac{x^3}{3}+\frac{x^5}{5}-\dots \dots .+{(-1)}^n\frac{x^{2n+1}}{2n+1}+\dots \dots$ differentiating $f(x)$ three times with respect to $x$ .

  $f^{‴}(x)=\frac{d^3}{d{x}^3}(f(x))=-\frac{6}{3}+\frac{(\mathrm{3.4.5})x^2}{5}-\dots \dots +{(-1)}^n\frac{[(2n+1)(2n)(2n-1)]x^{2n-2}}{2n+1}+\dots \dots$

Therefore, $f^{‴}(0)=-\frac{1}{2}$