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11Alexandr11 [23.1K]
3 years ago
6

Determine the intervals on which the function is concave up or concave down. (Enter your answers using interval notation. If an

answer does not exist, enter DNE.)
f(θ) = 15 θ + 15 sin2(θ), [0, π]

Mathematics
2 answers:
Vaselesa [24]3 years ago
8 0

Answer:

Step-by-step explanation:

attached is the solution

BabaBlast [244]3 years ago
8 0

Answer:

The function is concave up on the interval (0, π/4]

And concave down on the interval [-π/4, 0)

Step-by-step explanation:

To investigate if a function is concave up or concave down, we investigate the second derivative of the function.

Given f(θ) = 15θ + 15sin²θ

Let us differentiate this function twice in succession.

f'(θ) = 15 + 15(2sinθcosθ) = 15 + 15sin2θ

f''(θ) = 30sin2θ.

The function is concave upward when it's second derivative is greater than zero. That is, when

f''(θ) > 0

=> 30sin2θ > 0

=> sin2θ > 0

=> 0 < θ ≤ π/4

The interval is (0, π/4]

The function is concave down when it's second derivative is less than zero. That is when

f''(θ) < 0

=> 30sin2θ < 0

=> sin2θ < 0

=> -π/4 ≤ θ < 0

The interval is [-π/4, 0)

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The probability that any postcard posted in Portugal on Monday is delivered to the UK within a week is 0.62.
sashaice [31]

Answer:

You can expect at least 9 of her postcards to arrive within a week

To find that, we multiply the number of postcards by the probability

15 x 0.62 = 9.3

Hope this helps

Good Luck

5 0
3 years ago
25 to 75 in ratio lowest term
ANTONII [103]
25 to 75 in lowest terms:

25:75 = 1:3
8 0
3 years ago
Write an equation of the line that passes through (-4, -1) and is perpendicular to the line y= 4/3x+6.
scoundrel [369]

Answer:

y = (-3/4)x - 4

Step-by-step explanation:

The slopes of perpendicular lines are opposite reciprocals of each other. In other words, if the slope of one line is a/b, then the slope of the line perpendicular to it would be -b/a.

Here, the given slope is 4/3, so the slope of the perpendicular line is -3/4.

We are given a point (-4, -1) and we know the slope, so we can find the point-slope form of the line. Point-slope form is written as y - y1 = m(x - x1), where (x1,y1) is the point and m is the slope. Here, x1 = -4 and y1 = -1 and m = -3/4. So:

y - y1 = m(x - x1)

y - (-1) = (-3/4) * (x - (-4)) = (-3/4) * (x + 4)

y + 1 = (-3/4)x + (-3/4) * 4 = (-3/4)x - 3

y = (-3/4)x - 3 - 1

y = (-3/4)x - 4

3 0
3 years ago
What is the gradient of a line<br> perpendicular to a line with a gradient<br> of 5
ZanzabumX [31]

Answer:

-1/5

Step-by-step explanation:

When a line is perpendicular to another line, it's gradient will be a reciprocal.

To calculate the reciprocal of a number you have to flip the fraction the other way around.

So, in the case our current gradient is 5/1 (5/1 is the same as just 5) and we need to flip in over to get 1/5, which is our new gradient.

The final step is to make the number negative because the line is going down. This gives us the final answer of -1/5.

3 0
2 years ago
Find the slope-intercept equation of the line that passes through (0,-1) and has a slope of -2/3 .
Rom4ik [11]

Answer:

y= -2/3x-1

Step-by-step explanation:

Hope this helps!

5 0
3 years ago
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