Mathematics: Find the absolute maximum and absolute minimum values of f on the given interval. f(t) = 2 cos(t) + sin(2t),[0, π/2].

Find the absolute maximum and absolute minimum

Ответ:

tmrsavage02p7cj16

02.02.2022

Mathematics

The absolute maximum is $\frac{3\sqrt 3}2$ and the absolute minimum value is

Step-by-step explanation:

Differentiate of both sides w.r.t. ,

$f(t)=2 \cos t+\sin 2t$

$\Rightarrow f'(t)=-2\sin t+2\cos 2t$

Now take f'(t)=0

$\Rightarrow -2\sin t+2\cos 2t=0$

$\Rightarrow 2\cos 2t=2\sin t$

$\Rightarrow \cos 2t=\sin t$

$\Rightarrow 1-2\sin ^2t =\sin t \quad \quad [\because \cos 2t = 1-2\sin ^2t]$

$\Rightarrow 2\sin ^2t+\sin t-1=0$

$\Rightarrow 2\sin ^2t+2\sin t-\sin t-1=0$

$\Rightarrow 2\sin t(\sin t+1)-1(\sin t+1)=0$

$\Rightarrow (\sin t+1)(2\sin t-1)=0$

$\Rightarrow \sin t+1=0 \;\text{and}\; 2\sin t-1=0$

$\Rightarrow \sin t =-1 \;\text{and}\; \sin t =\frac 12$

In the interval $0\leq t\leq \frac {\pi}2$ , the answer to this problem is $\frac {\pi}6$

Now find the second derivative of f(t) w.r.t. ,

$f''(t)=-2\cos t-4\sin 2t$

$\Rightarrow \left[f''(t)\right]_{t=\frac {\pi}6}=-2\times \frac {\sqrt 3}2-4\times \frac{\sqrt 3}2=-3\sqrt 3$

Thus, f(t) is maximum at $t=\frac {\pi}6$ and minimum at t=0

$\left[f(t)\right]_{t=\frac {\pi}6}=2\times \frac {\sqrt 3}2+\frac{\sqrt 3}2=\frac{3\sqrt 3}2\;\text{and}\;\left[f(t)\right]_{t=\frac{\pi}2}= 2\times 0+0=0$

Hence, the absolute maximum is $\frac{3\sqrt 3}2$ and the absolute minimum value is .

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Ответ:

javlo4461

16.02.2023

Mathematics

So if I understand this correctly, this would look something like this:

A first number

let's call it x

plus twice a second number

let's call the second number y:

x+2y=5

is 5.

this means that x=5-2y!

twice the first number plus the second totals 31.

2x+y=31

from the last one, i know that y=31-2x

and i can insert it into the first equation:

So i know that x=5-2y

and that

2x+y=31
i will substitute x here

2(5-2y)+y=31

remove the bracket

10-4y+y=31

10-3y=31

remove 10 from both sides

-3y=21

then

y=-7

and
x=5-2(-7)=5+14=19

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Find the absolute maximum and absolute minimum values of f on the given interval. f(t) = 2 cos(t) + sin(2t),[0, π/2].

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