Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f.
18. f(x) = 1/x , 1 < x < 3
I can graph it just fine, but for my answer I got that the function has none of the above. Is this correct?
you are correct. It has a vertical asymptote at x=0 and shoots off up and down forever
To find the absolute and local maximum and minimum values of the function f(x) = 1/x on the interval 1 < x < 3, we can start by sketching the graph:
Since f(x) = 1/x, we can start by plotting a few key points:
When x = 1, f(1) = 1/1 = 1.
When x = 2, f(2) = 1/2 = 0.5.
When x = 3, f(3) = 1/3 ≈ 0.333.
Next, we can determine the behavior of the graph as x approaches 1 and 3 separately.
As x approaches 1 from the right (∞), f(x) approaches -∞. As x approaches 1 from the left (0), f(x) approaches ∞. This means there is a vertical asymptote at x = 1 and the function is not defined at x = 1.
As x approaches 3 from the right (∞), f(x) approaches 1/3 ≈ 0.333. As x approaches 3 from the left (0), f(x) approaches ∞. This means there is a horizontal asymptote at y = 0 and a vertical asymptote at x = 3.
The graph of the function f(x) looks like a hyperbola opening downwards and is undefined at x = 1. It approaches the x-axis as x approaches positive infinity and approaches y = 0 as x approaches 3.
Considering that the function has a vertical asymptote at x = 1, as well as an undefined value at that point, there are no absolute or local maximum or minimum values for this function on the interval 1 < x < 3.
To sketch the graph of f(x) = 1/x, 1 < x < 3, you can follow these steps:
1. Determine the vertical asymptotes: Since the denominator cannot be zero, x = 0 is a vertical asymptote of the graph.
2. Determine the behavior near the vertical asymptotes: As x approaches 0 from the positive side, f(x) approaches positive infinity, and as x approaches 0 from the negative side, f(x) approaches negative infinity.
3. Determine the behavior as x approaches positive and negative infinity: As x approaches positive or negative infinity, f(x) approaches zero.
4. Find the x-intercept and y-intercept: For the x-intercept, set f(x) = 0 and solve for x. For the y-intercept, plug in x = 0 and solve for f(x).
Based on these steps, you can sketch the graph of f(x) = 1/x, 1 < x < 3. The graph should resemble a hyperbola opening upward and downward.
Now let's analyze the absolute and local maximum and minimum values of f(x) within the given interval 1 < x < 3.
Since f(x) = 1/x is a decreasing function on this interval, it does not have any local maximum or minimum values. However, it does have an absolute maximum and minimum.
To find the absolute maximum and minimum values, we need to consider the behavior of f(x) at the interval endpoints.
When x = 1, f(1) = 1/1 = 1.
When x = 3, f(3) = 1/3.
Therefore, the absolute maximum value of f(x) is 1 at x = 1, and the absolute minimum value is 1/3 at x = 3.
In conclusion, f(x) = 1/x, 1 < x < 3, has an absolute maximum value of 1 at x = 1, and an absolute minimum value of 1/3 at x = 3. It does not have any local maximum or minimum values within this interval.