Could sin ⁡ , cos ⁡ \sin, \cos as a couple hurt?

Calculus Level 3

f ( x ) = d d x sin ⁡ ( cos ⁡ ( x ) ) \large f(x) = \frac d{dx} \sin(\cos(x))

Find the difference between maximum and minimum value of f ( x ) f(x) .


The answer is 2.

Viki Zeta by Viki Zeta , 19, India

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1 solution

Chew-Seong Cheong
Oct 19, 2016

f ( x ) = d d x sin ⁡ ( cos ⁡ x ) = − sin ⁡ x cos ⁡ ( cos ⁡ x ) ⟹ f m a x ( x ) = − sin ⁡ ( − π 2 ) cos ⁡ ( cos ⁡ ( − π 2 ) ) = 1 ⟹ f m i n ( x ) = − sin ⁡ ( π 2 ) cos ⁡ ( cos ⁡ ( π 2 ) ) = − 1 ⟹ f m a x ( x ) − f m i n ( x ) = 2 \begin{aligned} f(x) & = \frac d{dx} \sin (\cos x) \\ & = -\sin x \cos (\cos x) \\ \implies f_{max}(x) & = - \sin \left( -\frac \pi 2 \right) \cos \left( \cos \left( -\frac \pi 2 \right) \right) = 1 \\ \implies f_{min}(x) & = - \sin \left(\frac \pi 2 \right) \cos \left( \cos \left(\frac \pi 2 \right) \right) = -1 \\ \implies f_{max}(x) - f_{min}(x) & = \boxed{2} \end{aligned}

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