Mary Reyes

2021-12-31

Find range of the function

$f\left(x\right)=3\left|\mathrm{sin}x\right|-4\left|\mathrm{cos}x\right|$

I tried to do by using the trigonometric identities

$\mathrm{sin}}^{2}x=\frac{1-\mathrm{cos}2x}{2};\text{}\text{}{\mathrm{cos}}^{2}x=\frac{1+\mathrm{cos}2x}{2$

So

$f\left(x\right)=3\sqrt{\frac{1-\mathrm{cos}2x}{2}}-4\sqrt{\frac{1+\mathrm{cos}2x}{2}}$

but don't know how to proceed from here

I tried to do by using the trigonometric identities

So

but don't know how to proceed from here

Kayla Kline

Beginner2022-01-01Added 37 answers

for all

Mollie Nash

Beginner2022-01-02Added 33 answers

Let

$f\left(x\right)=3\left|\mathrm{sin}x\right|-4\left|\mathrm{cos}x\right|$

It's easy to show f(x) is periodic with$T=\pi$ and f(x) is even. So it's enough to consider $0\le x\le \frac{\pi}{2}$

If$0\le x\le \frac{\pi}{2}$ then

$0\le \mathrm{sin}x\le 1\Rightarrow 0\le \left|\mathrm{sin}x\right|\le 1$

And

$0\le \mathrm{cos}x\le 1\Rightarrow 0\le \left|\mathrm{cos}x\right|\le 1$

Therefore we have$-4\le f\left(x\right)\le 3$ but this could be only an upper bound not the actual range.

If$0\le x\le \frac{\pi}{2}$ then

$\left(x\right)=3\mathrm{sin}x-4\mathrm{cos}x\Rightarrow f\prime \left(x\right)=3\mathrm{cos}x+4\mathrm{sin}x>0\Rightarrow$

$f\left(0\right)\le f\left(x\right)\le f\left(\frac{\pi}{2}\right)\Rightarrow -4\le f\left(x\right)\le 3$

Confirming the first result. Actually because f(x) is an increasing function in that interval, the results are the same. So we can conclude that$-4\le f\left(x\right)\le 3$ and WA verifies that.

It's easy to show f(x) is periodic with

If

And

Therefore we have

If

Confirming the first result. Actually because f(x) is an increasing function in that interval, the results are the same. So we can conclude that

Vasquez

Expert2022-01-09Added 669 answers

Find an equation of the plane. The plane through the points (2, 1, 2), (3, −8, 6), and (−2, −3, 1), help please

A consumer in a grocery store pushes a cart with a force of 35 N directed at an angle of $25}^{\circ$ below the horizontal. The force is just enough to overcome various frictional forces, so the cart moves at a steady pace. Find the work done by the shopper as she moves down a $50.0-m$ length aisle.

??What is the derivative of $\mathrm{arcsin}\left[{x}^{\frac{1}{2}}\right]$?

What is the derivative of $y=\mathrm{arcsin}\left(\frac{3x}{4}\right)$?

Determine if the graph is symmetric about the $x$-axis, the $y$-axis, or the origin.$r=4\mathrm{cos}3\theta $.

How to differentiate $1+{\mathrm{cos}}^{2}\left(x\right)$?

What is the domain and range of $\left|\mathrm{cos}x\right|$?

How to find the value of $\mathrm{csc}74$?

How to evaluate $\mathrm{sec}\left(\pi \right)$?

Using suitable identity solve (0.99)raised to the power 2.

How to find the derivative of $y=\mathrm{tan}\left(3x\right)$?

Find the point (x,y) on the unit circle that corresponds to the real number t=pi/4

How to differentiate ${\mathrm{sin}}^{3}x$?

A,B,C are three angles of triangle. If A -B=15, B-C=30. Find A , B, C.

Find the value of $\mathrm{sin}{270}^{\circ}$.