I also have these other four question I need help on please!!

1. The graph of y = f ′(x), the derivative of f(x), is shown below. Given f(4) = 6, evaluate f(0).

A. -2
B. 2
C. 4 <--- My choice
D. 10

2. The graph of f ′(x) is continuous and decreasing with an x-intercept at x = 0. Which of the following statements is false?

A. The graph of f has an inflection point at x = 0.
B. The graph of f has a relative maximum at x = 0.
C. The graph of f is always concave down.
D. The graph of the second derivative is always negative.

3. f is a differentiable function on the interval [0, 1] and g(x) = f(4x). The table below gives values of f '(x). What is the value of g '(0.1)?

x 0.1 0.2 0.3 0.4 0.5
f '(x) 1 2 3 –4 5

A. -16 <----- My choice
B. -4
C. 4
D. Cannot Be Determined

4. The table of values below shows the rate of water consumption in gallons per hour at selected time intervals from t = 0 to t = 12.

Using a left Riemann sum with 5 subintervals, estimate the total amount of water consumed in that time interval.

x 0 2 5 7 11 12
f(x) 5.7 5.0 2.0 1.2 0.6 0.4

A. 22.1
B. 34.8
C. 35.8
D. None of these

THANK YOU!!! :)

Anybody know how I can provide the image/ link for question 1 without being blocked by Jiskha?

Problem 1 shows a specific graph for the problem.

I reworked my process for number 1 and I got B. 2 this time. Is that correct?

#1. No idea

#2. A - consider y = -x^2
#3. Looks good
#4
x 0 2 5 7 11 12
f(x) 5.7 5.0 2.0 1.2 0.6 0.4
the left sum is, of course,
5.7*2 + 5.0*3 + 2.0*2 + 1.2*4 + 0.6*1 = 35.8

It is 2

1. To evaluate f(0), we need to find the value of f(x) at x=0. Given that f'(x) is the derivative of f(x), we can use the graph of f'(x) to find the value of f(x) at any point.

Looking at the graph, we can see that f'(x) is positive and increasing from x=0 to x=4. This means that f(x) is also increasing in this interval. Since f(4) = 6, we know that f(x) has a value of 6 at x=4.

To find f(0), we need to determine the change in f(x) from x=4 to x=0. Since f(x) is increasing, we can subtract the value of f'(x) at x=0 from the value of f(x) at x=4. From the graph, we can see that f'(x) at x=0 is equal to 2. Therefore, we can find f(0) by subtracting 2 from 6: f(0) = 6 - 2 = 4.

So the correct answer is C. 4.

2. The graph of f'(x) tells us about the behavior of the derivative of f(x), but we need to consider the properties of f(x) itself.

Since f'(x) is continuous and decreasing, f(x) is a function that is decreasing at a decreasing rate. Since f'(x) has an x-intercept at x = 0, this means that f(x) has a relative minimum at x = 0.

Therefore, the false statement is B. The graph of f has a relative maximum at x = 0.

3. To find the value of g'(0.1), we can use the chain rule. Since g(x) = f(4x), we can rewrite g'(x) as f'(4x) * 4.

From the table, we can find that f'(x) is equal to 1 at x=0.1. Therefore, f'(4x) is equal to f'(0.4) which is -4.

Now, using the chain rule, g'(0.1) is calculated as f'(4(0.1)) * 4 = f'(0.4) * 4 = -4 * 4 = -16.

So the correct answer is A. -16.

4. To estimate the total amount of water consumed in the time interval using a left Riemann sum, we need to calculate the sum of the areas of the rectangles formed by the function values and the width of each subinterval.

The width of each subinterval is (12 - 0)/5 = 2.4, since there are 5 subintervals. We can calculate the sum of the areas by multiplying each function value of f(x) by the width and adding them up:

(5.7 * 2.4) + (5.0 * 2.4) + (2.0 * 2.4) + (1.2 * 2.4) + (0.6 * 2.4) = 13.68 + 12 + 4.8 + 2.88 + 1.44 = 34.8

So the estimated total amount of water consumed in that time interval is 34.8.

Therefore, the correct answer is B. 34.8.