Algebraic and Trigonometric expression is &gt;0 for all real x2x2sin⁡x+2xcos⁡x+2x2+1

Step 1For sin⁡x+1&gt;0 we obtain:2x2sin⁡x+2xcos⁡x+2x2+1=2(1+sin⁡{x})x2+2xcos⁡{x}+1≥0BecauseΔ4=cos2x−2(1+sin⁡{x})=−(1+sin⁡{x})2&lt;0.If sin⁡{x}+1=0 so2x2sin⁡x+2xcos⁡x+2x2+1=1&gt;0.

Step 1By completing the square,2(sin⁡x+1)x2+2cos⁡x,x+1=2(sin⁡x+1)(x+cos⁡x2(sin⁡x+1))2+1−cos2x2(sin⁡x+1)=2(sin⁡x+1)((x+cos⁡x2(sin⁡x+1))2+sin⁡x+12)Obviously, sin⁡x+1≥0. In case of equality, sin⁡x=−1⇒cos⁡x=0 and the expression reduces to 1.

Step 1For sin⁡x+1&gt;0 we obtain:2x2sin⁡x+2xcos⁡x+2x2+1=2(1+sin⁡{x})x2+2xcos⁡{x}+1≥0BecauseΔ4=cos2x−2(1+sin⁡{x})=−(1+sin⁡{x})2&lt;0.If sin⁡{x}+1=0 so2x2sin⁡x+2xcos⁡x+2x2+1=1&gt;0.

Step 1By completing the square,2(sin⁡x+1)x2+2cos⁡x,x+1=2(sin⁡x+1)(x+cos⁡x2(sin⁡x+1))2+1−cos2x2(sin⁡x+1)=2(sin⁡x+1)((x+cos⁡x2(sin⁡x+1))2+sin⁡x+12)Obviously, sin⁡x+1≥0. In case of equality, sin⁡x=−1⇒cos⁡x=0 and the expression reduces to 1.

"Algebraic and Trigonometric expression is >0 for all..." was answered by students like you

High School
Algebra
Algebra I
Quadratic function and equation

maxime99bl6

Answered question

2022-03-18

Algebraic and Trigonometric expression is

> 0

for all real x

2 x^{2} \sin x + 2 x \cos x + 2 x^{2} + 1

Answer & Explanation

eslasadadsc

Beginner2022-03-19Added 7 answers

Step 1
For

\sin x + 1 > 0

we obtain:

2 x^{2} \sin x + 2 x \cos x + 2 x^{2} + 1 = 2 (1 + \sin {x}) x^{2} + 2 x \cos {x} + 1 \geq 0

Because

\frac{Δ}{4} = \cos^{2} x - 2 (1 + \sin {x}) = - {(1 + \sin {x})}^{2} < 0 .

\sin {x} + 1 = 0

2 x^{2} \sin x + 2 x \cos x + 2 x^{2} + 1 = 1 > 0 .

Keith Steele

Beginner2022-03-20Added 2 answers

Step 1
By completing the square,

2 (\sin x + 1) x^{2} + 2 \cos x, x + 1

= 2 (\sin x + 1) {(x + \frac{\cos x}{2 (\sin x + 1)})}^{2} + 1 - \frac{\cos^{2} x}{2 (\sin x + 1)}

= 2 (\sin x + 1) ({(x + \frac{\cos x}{2 (\sin x + 1)})}^{2} + \frac{\sin x + 1}{2})

Obviously,

\sin x + 1 \geq 0

. In case of equality,

\sin x = - 1 \Rightarrow \cos x = 0

and the expression reduces to 1.

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Algebra I

Find the volume V of the described solid S
A cap of a sphere with radius r and height h.
V=??
Whether each of these functions is a bijection from R to R.
a) $f (x) = - 3 x + 4$
b) $f (x) = - 3 x^{2} + 7$
c) $f (x) = \frac{x + 1}{x + 2}$
$d) f (x) = x^{5} + 1$
?
In how many different orders can five runners finish a race if no ties are allowed???
State which of the following are linear functions?
a. $f (x) = 3$
b. $g (x) = 5 - 2 x$
c. $h (x) = \frac{2}{x} + 3$
d. $t (x) = 5 (x - 2)$
Three ounces of cinnamon costs $2.40. If there are 16 ounces in 1 pound, how much does cinnamon cost per pound?
A square is also a
A)Rhombus;
B)Parallelogram;
C)Kite;
D)none of these
What is the order of the numbers from least to greatest.
$A = 1.5 \times 10^{3}$ ,
$B = 1.4 \times 10^{- 1}$ ,
$C = 2 \times 10^{3}$ ,
$D = 1.4 \times 10^{- 2}$
Write the numerical value of $1.75 \times 10^{- 3}$
Solve for y. 2y - 3 = 9
A)5;
B)4;
C)6;
D)3
How to graph $y = \frac{1}{2} x - 1$ ?
How to graph $y = 2 x + 1$ using a table?
simplify $\sqrt{257}$
How to find the vertex of the parabola by completing the square $x^{2} - 6 x + 8 = y$ ?
There are 60 minutes in an hour. How many minutes are there in a day (24 hours)?
Write 18 thousand in scientific notation.

Didn't find what you were looking for?