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An instantaneous displacement of a simple harmonic oscillator is $x=A \cos (\omega t+\pi / 4)$. Its speed will be maximum at time
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$\pi / 4 \omega$
Velocity, $v=\frac{d x}{d t}=-A \omega \sin (\omega t+\pi / 4)$
Velocity will be maximum, when $\omega t+\pi / 4=\pi / 2$ or $\omega t=\pi / 2-\pi / 4=\pi / 4$ or $t=\pi / 4 \omega$
Velocity will be maximum, when $\omega t+\pi / 4=\pi / 2$ or $\omega t=\pi / 2-\pi / 4=\pi / 4$ or $t=\pi / 4 \omega$
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