Question

Which of the following is true

(a) Value of \(\frac{d^m (Sin⁡(nx))}{dx^m}\) is always positive for m=0, 1, 4, 5, 8, 9… for 0 < nx < ^π⁄2 and n<0

(b) Value of \(\frac{d^m (Sin⁡(nx))}{dx^m}\) is always positive for m=2, 3, 6, 7, 10, 11… for 0 < nx < ^π⁄2 and n>0

(c) Value of \(\frac{d^m (Sin⁡(nx))}{dx^m}\) is always positive for m=0, 1, 4, 5, 8, 9… for 0 < nx < ^π⁄2 and n>0

(d) Value of \(\frac{d^m (Sin⁡(nx))}{dx^m}\) is always positive for m=2, 3, 6, 7, 10, 11… for 0 < nx < ^π⁄2 and n<0

I have been asked this question in an internship interview.

This is a very interesting question from The nth Derivative of Some Elementary Functions topic in portion Differential Calculus of Engineering Mathematics

Answer 1

Right answer is (c) Value of \(\frac{d^m (Sin⁡(nx))}{dx^m}\) is always positive for m=0, 1, 4, 5, 8, 9… for 0 < nx < ^π⁄2 and n>0

Explanation: Here,

\(\frac{d(Sin⁡(nx))}{dx} = n Cos(nx)\) …………….(m=1)

\(\frac{d^2 (Sin⁡(nx))}{dx^2} = -n^2 Sin(nx)\) …..(m=2)

\(\frac{d^3 (Sin⁡(nx))}{dx^3} = -n^3 Cos(nx)\) …..(m=3)

\(\frac{d^4 (Sin⁡(nx))}{dx^4} = n^4 Sin(nx)\) ……(m=4)

So the value of \(\frac{d^m (Sin⁡(nx))}{dx^m} = \begin{cases}n^m Cos(nx) \,\,\, m=1,5,9,….\\-n^m Sin(nx) \,\,\, m=2,6,10…\\-n^m Cos(nx)\,\,\,m=3,7,11…..\\n^m Sin(nx)\,\,\,m=4,8,12….\end{cases} \)

Hence, for n>0 and 0<nx<(π)/2 only \(\frac{d^m (Sin⁡(nx))}{dx^m}\) is positive only when m = 1,4,5,8,9,….. otherwise negative.