Answer:
(D) I and II only
Step-by-step explanation:
A function has a limit at a point if the limit from the left and the limit from the right are the same value.
Here f(3) = 5, and the linear function defined for x ≤ 3 is continuous, so the left limit exists.
The function defined for x > 3, when evaluated at x = 3, also has a value of 5. That function, too, is continuous up to x = 3, so the right limit is defined.
The left and right limits at x=3 are y=5, so the limit exists. (Statement I is true.)
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The function f(x) is defined as 5 for x=3, so the limit exists and is the same as the function value at x=3. That means the function is continuous at x=3. (You can draw the graph through that point without lifting your pencil.) (Statement II is true.)
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The slope of f(x) for x < 3 is 1; the slope for x > 3 is 4. There is a discontinuity in the slope at x=3, so the function is not differentiable there. (Statement III is false.)
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The graph shows the function in red and its derivative in blue. You will notice there is a discontinuity (step change) at x=3.
Answer:
Option A.
Step-by-step explanation:
Let M be the milk per gallon.
C be the cookies per dozen.
B be the bundle (one gallon of milk and a dozen cookies ).
Milk can be sold by itself for a profit of $1.50 per gallon. Cookies can likewise be sold at a profit of $2.50 per dozen and bundle is sold for a profit of $3.00 per bundle.
We need to maximize the profits. So, our objective function is
Therefore, the correct option is A.
Differentiate both sides, treating as a function of
. Let's take it one term at a time.
Power, product and chain rules:
Product and chain rules:
Product and chain rules:
The derivative of 0 is, of course, 0. So we have, upon differentiating everything,
Isolate the derivative, and solve for it:
(See comment below; all the 6s should be 2s)
We can simplify this a bit by multiplying the numerator and denominator by to get rid of that fraction in the denominator.