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lesya [120]
3 years ago
14

Determine the values of the constants B and C so that the function given below is differentiable.

Mathematics
1 answer:
laila [671]3 years ago
5 0
For the function to be differentiable, its derivative has to exist everywhere, which means the derivative itself must be continuous. Differentiating gives

f'(x)=\begin{cases}24x^2&\text{for }x1\end{cases}

The question mark is a placeholder, and if the derivative is to be continuous, then the question mark will have the same value as the limit as x\to1 from either side.

\displaystyle\lim_{x\to1^-}f'(x)=\lim_{x\to1}24x^2=24
\displaystyle\lim_{x\to1^+}f'(x)=\lim_{x\to1}B=B

So the derivative will be continuous as long as B=24

For the function to be differentiable everywhere, we need to require that f(x) is itself continuous, which means the following limits should be the same:

\displaystyle\lim_{x\to1^-}f(x)=\lim_{x\to1}8x^3=8
\displaystyle\lim_{x\to1^+}f(x)=\lim_{x\to1}Bx+C=24+C

24+C=8\implies C=-16

So, the function should be

f(x)=\begin{cases}8x^3&\text{for }x\le1\\24x-16&\text{for }x>1\end{cases}

with derivative

f'(x)=\begin{cases}24x^2&\text{for }x
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(Picture) MULTIPLYING MONOMIALS AND BINOMIALS
motikmotik

Answer:

Second option: 81y^4 - 16x^2, the difference of squares

Step-by-step explanation:

(9y^2-4x)(9y^2+4x) is a special product named difference of squares, then we can apply this formula:

(a-b)(a+b)=a^2-b^2, with a=9y^2 and b=4x, then:

(9y^2-4x)(9y^2+4x)=(9y^2)^2 - (4x)^2

(9y^2-4x)(9y^2+4x)=(9)^2 (y^2)^2 - (4)^2 (x)^2

(9y^2-4x)(9y^2+4x)=81y^(2*2) - 16x^2

(9y^2-4x)(9y^2+4x)=81y^4 - 16x^2


7 0
3 years ago
Chad drove 168 miles in 3 hours. How many miles per hour did chad drive? Chad will drive 672 more miles. He continues to drive a
larisa86 [58]

168/3 = 56

<em>Therefore, Chad is driving the car in 56 mph.</em>

672/56 = 12

<em>Therefore, Chad drives 672 miles in 12 hours.</em>

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<em>Therefore, Chad drives 28 miles per gallon of gas.</em>

672/28 = 24

<em>Therefore, Chad uses 24 gallons of gas to drive 672 miles.</em>

3 0
3 years ago
I need help please ​
devlian [24]
I am pretty sure its B (minimum at -1, -4) because that would be were the first point would be at. The Graphing of this equation would be a "upside down u" shape.
4 0
3 years ago
What does this mean
MissTica
That is basically splitting the sentence so that it’s easier for you to solve it. It’s basically the same thing
4 0
3 years ago
Which results only in a horizontal compression of y= 1/х by a factor of 6?
marta [7]

Answer:

I'm assuming you mean a compression of factor 6

In that case, it will be h = 1/6x

5 0
3 years ago
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