Suppose that g(x) is a function that is defined and differentiable for all real numbers x and that g(x) has the following properties:
(i) g(0)=5 and g'(0)=-5/6 .
(ii) g(7)=6 and g'(7)=6 .
(iii) g(x) is concave up for x<7 and concave down for x>7.
(iv) g(x)>_-15 for all x.
Use these properties to answer the following questions.
(a) How many zeros does g have?
(b) How many zeros does g have?
(a) Well, it's clear that g has at least one zero, since g(0) = 5 and we know g(x) crosses the x-axis somewhere between x = 0 and x = 7. But how many more zeros does it have? We don't know for sure, so let's assume it's a bit undecided. We'll call it a "zero maybe", just to play it safe.
(b) Ah, the same question again? Well, since g is defined and differentiable for all real numbers, we can say that g has infinitely many zeros! Think about it - for every x where g(x) = 0, it's also true that g'(x) = 0. So we have an infinite clown parade of zeros marching through the real number line. Hilarious, isn't it?
(a) We can use the properties (iii) and (iv) to conclude that the function g(x) is continuous for all x and does not cross the x-axis. This means that g(x) does not have any zeros.
(b) Since g(x) does not have any zeros (as determined in part (a)), g(x) does not have any zeros.
To answer these questions, we need to analyze the given properties of the function g(x).
(a) How many zeros does g have?
To find the number of zeros of g(x), we look for changes in sign or where g(x) crosses the x-axis. From the given properties, we have the following information:
(i) g(0) = 5, which means the function has a point on the positive y-axis at (0, 5).
(ii) g(7) = 6, which means the function has a point on the positive y-axis at (7, 6).
Since g(x) is a continuous function and concave up for x < 7 and concave down for x > 7, it implies that g(x) must cross the x-axis at least once between x = 0 and x = 7. This guarantees the existence of at least one zero.
However, we cannot definitively say if g(x) crosses the x-axis again after x = 7, as we don't have enough information about the behavior of the function beyond x = 7. Therefore, the number of zeros of g cannot be determined exactly based on the given information.
(b) How many zeros does g have?
Until we have certainty about the behavior of g(x) for x > 7, we cannot determine the exact number of zeros for g(x). It is possible that g(x) never crosses the x-axis again after x = 7, leaving only one zero. Alternatively, g(x) might cross the x-axis multiple times for x > 7, resulting in additional zeros.
To provide a more accurate answer, we would need additional information or conditions on the behavior of g(x) beyond x = 7.