How do I determine the time $t$ in which a function for the first time achieves the maximum and minimum value?

Question

So I have to use this function: $I(t) = 4.2 \times \sin (0.04 π t) + 1.2$ and somehow find the maximum and minimum value, but I'm struggling to understand how.

Answer 1

$I(t)=4.2\sin(0.04\pi t)+1.2$

$\implies I'(t) = 0.168\pi\cdot\cos(0.04\pi t)$

Set $I'(t) = 0$

$0.168\pi\cdot\cos(0.04\pi t) = 0$

$\implies \cos(0.04\pi t)= 0$

$\implies 0.04 \pi t = \frac\pi2,\frac{3\pi}2$

NOTE: I am considering $0\ge t\ge2\pi$

$\implies t = \frac1{0.08} $ or $t =\frac3{0.08}$

Now ;

$I''(t) = -0.00672\pi^2\sin(0.04\pi t)$

$I''(\frac1{0.08} )= -0.00672\pi^2\sin(\frac\pi2)$

$I''(\frac1{0.08} )= -0.00672\pi^2 \lt 0$

The second derivative is negative so the function has a maxima at $t = \frac1{0.08}$

$I''(\frac3{0.08}) =-0.00672\pi^2\sin(3\frac\pi2) $

$ I''(\frac3{0.08}) =0.00672\pi^2 \gt 0$

The second derivative is positive so the function has a minima at $t = \frac{3}{0.08}$