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White raven [17]
3 years ago
12

Find the values of m and b that make the following function differentiable.

Mathematics
1 answer:
KengaRu [80]3 years ago
7 0

f(x)=\begin{cases}x^2&\text{for }x\le2\\mx+b&\text{for }x>2\end{cases}

In order to be differentiable everywhere, f must first be continuous everywhere, which means the limits from either side as x\to2 must be the same and equal to f(2). By definition, f(2)=2^2=4, and

\displaystyle\lim_{x\to2^-}f(x)=\lim_{x\to2}x^2=4

\displaystyle\lim_{x\to2^+}f(x)=\lim_{x\to2}(mx+b)=2m+b

so we need to have 4m+b=4.

For f to be differentiable at x=2, the derivative needs to be continuous at x=2, i.e.

\displaystyle\lim_{x\to2^-}f'(x)=\lim_{x\to2^+}f'(x)

We then need to have

\displaystyle\lim_{x\to2}2x=\lim_{x\to2}m\implies\boxed{m=4}

Then

2m+b=4\implies\boxed{b=-4}

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PIT_PIT [208]

The solution to the equation is p = 1/3 and q = undefined

<h3>How to solve the equation?</h3>

The equation is given as:

p^2 - 2qp + 1/q = (p - 1/3)

The best way to solve the above equation is by the use of a graphing calculator i.e. graphically

However, it can be solved algebraically too (to some extent)

Recall that the equation is given as:

p^2 - 2qp + 1/q = (p - 1/3)

Split the equation

So, we have

p^2 - 2qp + 1/q = 0

p - 1/3 = 0

Solve for p in p - 1/3 = 0

p = 1/3

Substitute p = 1/3  in p^2 - 2qp + 1/q = 0

So, we have

(1/3)^2 - 2q(1/3) + 1/q = 0

This gives

1/9 - 2/3q + 1/q = 0

This gives

2/3q + 1/q = -1/9

Multiply though by q

So, we have

2/3q^2 + 1 = -1/9q

Multiply through by 9

6q^2 + 9 = -q

So, we have

6q^2 + q + 9 = 0

Using the graphing calculator, we have

q = undefined

Hence. the solution to the equation is p = 1/3 and q = undefined

Read more about equations at:

brainly.com/question/13763238

#SPJ1

5 0
2 years ago
James is twice as old as Hannah. Four years from now, the sum of their ages will be 41. How old are they now?
NemiM [27]
James is 24.5 and Hannah is 12.5.
4 0
2 years ago
Thursday<br>Every row and column must add<br>to 24. Use only 4 through 12.<br>be<br>on​
maw [93]

Answer:

12 two times or 4 eight times

7 0
3 years ago
Jessica and Emily went shopping at a local boutique. Jessica purchased 4 bracelets and a new.pair of earrings for $9.50. Emily p
Radda [10]
What’s the complete question
5 0
3 years ago
Please help me this is due! 7th grade math offering 20 points and might give brainly if I get more than one answer :)
Bezzdna [24]

Answer:

I'm pretty sure it's $98.10

Step-by-step explanation:

$90*.09=8.1

90+8.1=98.1

which can also be written as $98.10

3 0
2 years ago
Read 2 more answers
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