The distance between an arbitrary point on the surface and the origin is
Recall that for differentiable functions
and
, the composition
attains extrema at the same points that
does, so we can consider an augmented distance function
The Lagrangian would then be
We have partial derivatives
Set each partial derivative to 0 and solve the system to find the critical points.
From the second equation it follows that either
or
. In the first case we arrive at a contradiction (I'll leave establishing that to you). If
, then we have
This means
so that the points on the surface closest to the origin are
.
(2x+3y)⁴
1) let 2x = a and 3y = b
(a+b)⁴ = a⁴ + a³b + a²b² + ab³ + b⁴
Now let's find the coefficient of each factor using Pascal Triangle
0 | 1
1 | 1 1
2 | 1 2 1
3 | 1 3 3 1
4 | 1 4 6 4 1
0,1,2,3,4,.. represent the exponents of binomials
Since our binomial has a 4th exponents, the coefficients are respectively:
(1)a⁴ + (4)a³b + (6)a²b² + (4)ab³ + (1)b⁴
Now replace a and b by their real values in (1):
2⁴x⁴ +(4)8x³(3y) + (6)(2²x²)(3²y²) + (4)(2x)(3³y³) + (1)(3⁴)(y⁴)
16x⁴ + 96x³y + 216x²y² + 216xy³ + 81y⁴
Answer:
A and E are correct option.
Step-by-step explanation:
We are given two triangle congruent.
If two triangles are congruent then their corresponding sides and angles are equal.
In ΔTUV ≅ ΔWXY
Now we write all the congruent sides and angles
- ∠T=∠W
- ∠U=∠X
- ∠V=∠Y
- TU≅WX
- UV≅XY
- TV≅WY
Now we see all the given option.
∠Y=∠V (TRUE, Congruent part of congruence triangles)
∠W=∠T (TRUE, Congruent part of congruence triangles)
Thus, A and E are correct option.
Answer:
(2.4, -1.2)
Step-by-step explanation:
Start by moving the x and the y to the same side and moving the number across the equal sign in both equations. We should now have y-0.45x=-2.3 and 2y+4.2x=7.8. We can use the elimination method by multiplying the first equation by -2 to get -2y+0.9x=4.6 and 2y+4.2x=7.8. From there, add the two equations together, eliminating y (-2+2=0). We now have 5.1x=12.4; divide both sides by 5.1 to get x=2.4. Then, in any of the two equations, let's use y-0.45x=-2.3, substitute x with 2.4. Now we have y-1.08=-2.3. Add 1.08 to both sides to get y=-1.22; round that to the nearest tenth to get -1.2.
1.Since 5 is less than 8, borrow 1 from the next column to make 15.
2.Calculate 15 - 8, which is 7.
3.Calculate 3 - 3, which is 0.
4.Since 2 is less than 5, borrow 1 from the next column to make 12.
5.Calculate 12 - 5, which is 7.
6..<span>Calculate 9 - 2, which is 7.</span>
7.<span>Therefore, 102.45 - 25.38 = 77.07.</span>
