Question

On which of the following intervals is the function f given by f(x) = x¹⁰⁰ + sinx - 1 strictly decreasing.

Answer 1

\(f(x) = x^{100} + sin\,x - 1\)

\(f'(x) = 100\,x^{99} + cos\,x\)

For strictly increasing, we have

\(100\,x^{99} + cos\,x > 0\)

⇒ \(100\,x^{99} > - cos\,x \)

⇒ \(100\,x^{99} > -1\)   \((\because -1\le cos\,x\le 1)\)

⇒ \(x^{99} > \frac{-1}{100}\)

⇒ \(x> \left(\frac{-1}{100}\right)^{\frac1{99}}\)

⇒ \(x > -0.96 \) (approx)

Hence, for values greater than -0.96, the given function f(x) is strictly increasing.