WILL GIVE BRAINLY FOR ANSWER!! Please help with this question!!! Given the piecewise function: f(x) = 1/2x + 5, x > 2 6, x = 2 x + 4, x < 2 a. Write f' (f prime) as a piecewise function b. Determine if f is differentiable at x = 2. Give a reason for your answer. Photo is attatched.

Answer:

A)

[tex]f'(x) = \left\{ \begin{array}{lIl} \frac{1}{2} & \quad x >2 \\ 0& \quad x =2\\1&\quad x<2 \end{array} \right.[/tex]

B) Continuous but not differentiable.

Step-by-step explanation:

So we have the piecewise function:

[tex]f(x) = \left\{ \begin{array}{lIl} \frac{1}{2}x+5 & \quad x >2 \\ 6& \quad x =2\\x +4&\quad x<2 \end{array} \right.[/tex]

A)

To write the differentiated piecewise function, let's differentiate each equation separately. Thus:

1)

[tex]\frac{d}{dx}[\frac{1}{2}x+5}][/tex]

Expand:

[tex]\frac{d}{dx}[\frac{1}{2}x]+\frac{d}{dx}[5][/tex]

The derivative of a linear equation is just the slope. The derivative of a constant is 0. Thus:

[tex]\frac{d}{dx}[\frac{1}{2}x+5}]=\frac{1}{2}[/tex]

2)

[tex]\frac{d}{dx}[6][/tex]

Again, the derivative of a constant is 0. Thus:

[tex]\frac{d}{dx}[6]=0[/tex]

3)

We have:

[tex]\frac{d}{dx}[x+4][/tex]

Expand:

[tex]\frac{d}{dx}[x]+\frac{d}{dx}[4][/tex]

Simplify:

[tex]=1[/tex]

Now, let's substitute our original equations for the differentiated equations. The inequalities will stay the same. Therefore:

[tex]f'(x) = \left\{ \begin{array}{lIl} \frac{1}{2} & \quad x >2 \\ 0& \quad x =2\\1&\quad x<2 \end{array} \right.[/tex]

B)

For a function to be differentiable at a point, the function must be a) continuous at that point, and b) the left and right hand derivatives must be equivalent.

Let's first determine if the function is continuous at the point. Remember that a function is continuous at a point if and only if:

[tex]\lim_{x \to n^-} f(n)= \lim_{x \to n^+}f(n)=f(n)[/tex]

Let's find the left hand limit of f(x) at it approaches 2.

[tex]\lim_{x \to 2^-}f(x)[/tex]

Since it's coming from the left, let's use the third equation:

[tex]\lim_{x \to 2^-}f(x)\\=\lim_{x \to 2^-}(x+4)[/tex]

Direct substitution:

[tex]=(2+4)=6[/tex]

So:

[tex]\lim_{x \to 2^-}f(x)=6[/tex]

Now, let's find the right-hand limit:

[tex]\lim_{x \to 2^+}f(x)[/tex]

Since we're coming from the right, let's use the first equation:

[tex]\lim_{x \to 2^+}(\frac{1}{2}x+5)[/tex]

Direct substitution:

[tex](\frac{1}{2}(2)+5)[/tex]

Multiply and add:

[tex]=6[/tex]

So, both the left and right hand limits are equivalent. Now, find the limit at x=2.  

From the piecewise function, we can see that the value of f(2) is 6.

Therefore, the function is continuous at x=2.

Now, let's determine differentiability at x=2.

For a function to be differentiable at a point, both the right hand and left hand derivatives must be equivalent.

So, let's find the derivative of the function as x approahces 2 from the left and from the right.

From the differentiated piecewise function, we can see that as x approaches 2 from the left, the derivative is 1.

As x approaches 2 from the right, the derivative is 1/2.

Therefore, the right and left hand derivatives are not the same.

Thus, the function is continuous but not differentiable.

Nidhi is creating a rectangular garden in her back yard. The length of the garden is 10 feet. The perimeter of the garden must be at least 40 feet and no more than 76 feet. Write and solve a compound inequality to find the range of values for the width, w, of the garden.

Prove that root 7 is an irrational number.

solve for n


8+8+n-4=3n-2n

The Cobb family’s natural gas bill was 85.50 for 95 cubic feet of gas, What is the unit rate? (Cost per cubic foot of gas?)

malik's solution to the equation [tex]\frac{2}{5}[/tex]×-4y=10

find the mode in the set 3,5,7,13,7,9,3
​

Wyatt owes $472.22 on his credit card. He returns two items, one for $49.63 and the other for $47.58.
Then he makes purchases of $39.57 and $32.21. What is the new balance on the card?

x+2/2×>1/2 in rational in equalities​

(-2.8)^6 (Round to three decimal places as needed)

How many years old are you have lived 1 billion second

70% of 120 = 120x70/100
How do you figure this out

Which term has a degree of 8? a.) 9xy^8 b.)-2x^3y^5 c.) 3x^2y^4 d.) 8xy

write a sentence as an equation


b is 380 more than the quantity y time 268​

Lena has $12 she bought a pack of craft paper for six dollars and a stapler for two dollars which expression correctly shows the money Lena has left

please me asap :)) will give 40 points
i only need 4-101

One year Hank had the lowest ERA (earned-run average, mean number of runs yielded per nine innings pitched) of any male pitcher at his school, with an ERA of
3.05. Also, Sally had the lowest ERA of any female pitcher at the school with an ERA of 2.96. For the males, the mean ERA was 3.848 and the standard deviation was
0.742. For the females, the mean ERA was 5.007 and the standard deviation was 0.722. Find their respective Z-scores. Which player had the better year relative to
their peers, Hank or Sally? (Note: In general, the lower the ERA, the better the pitcher.)
Tests
Hank had an ERA with a z-score of
Sally had an ERA with a z-score of

Are these the correct answers? Why or why not?

I need help with this ASAP

What is the value of x & y?

if a gymnasium is 80 feet long and 60 feet wide, what will the distance be between opposite corners
​

solve for x: -2(x+3)= -2 - 6

0

3

all real numbers

no solution

Identify the sequence below as arithmetic, geometric, or neither. Then find the next two terms a5 and a6
10, 30, 90, 270,

Hello what is 50+3 ? c h a t w i t h p e p s

What part of 20 is 25?

calculate the length of the base of a parallelogram if the area is 135cm and it's height is 9cm
​

21. Chandra is retiling her bathroom floor. Each tile is in the shape of a
square, and is 6.5 inches long. How many square inches does one tile
cover? *
WILL MARK AS BRAINLIEST

Sam can prepare 30 wall paintings in four weeks.How much time will he need to do 90 wall painting s​

Mr.rodriguez invests half money in land a tenth in stocks and a twentieth in bonds he puts the remaining 35,000 in his savings account what is the total amount of money that Mr Rodriguez saves and invests
​

I need help please no one helps me

\dfrac{4}{9},-\dfrac{4}{3},4, 9 4 ​ ,− 3 4 ​ ,4,