I'm having a lot of trouble with this problem:
Sketch the graph and show all local extrema and inflections.
f(x)= (x^(1/3)) ((x^2)-175)
I graphed the function on my graphing calculator and found the shape.
I also found the first derivative:
(7/3)(x^(4/3)) - (175/3)(x(-2/3))
But the number I found for x (when I set the deriv. to 0 to find the critical points) was 1.71, but this doesn't correspond to a min or a max. Then I plugged the equation of the first derivative in onto my calculator, and the corresponding y value of 1.71 was NOT zero.... furthermore, it seemed like there was a horizontal asymptote at 0, because the derivative never was 0.
I couldn't find an inflection point, because the second derivative never equaled zero.
Does this seem right-- that there is no local max, min, OR inflection point? If this is wrong, can you help me find the right values? I sketched the graph on my paper but I still haven't found the min, max, or inflection points.
I had the same first derivative as you did
y' = (7/3)(x^(4/3)) - (175/3)(x(-2/3))
= (1/3)x^(-2/3)[7x^2 - 175] by factoring
setting this equal to zero, ...
the first factor of (1/3)x^(-2/3) yields no solution but
7x^2-175=0 gives me
x = ± 5
my second derivative was (28/9)x^(1/3) + (350/9)x^(-5/3) which when set to zero has no answer.
so according to the math, there should be 2 points of max/min, but no point of inflection.
Does that fit in with your sketch?
I did not sketch the graph.
Let's go step by step to analyze the function and find the local extrema and inflection points.
Step 1: Finding critical points.
To find critical points, we need to set the first derivative equal to zero and solve for x.
The first derivative you calculated is:
f'(x) = (7/3)(x^(4/3)) - (175/3)(x^(-2/3))
Setting this equal to zero, we get:
(7/3)(x^(4/3)) - (175/3)(x^(-2/3)) = 0
To simplify this equation, we can multiply through by 3 to get rid of the fractions:
7(x^(4/3)) - 175(x^(-2/3)) = 0
Now we can solve for x. Unfortunately, it is not easy to solve this equation algebraically. One way to approach this is to use numerical methods such as graphing calculator or software to find the approximate values of x.
Based on your calculation, you found x ≈ 1.71. However, it seems like you may have made an error in your calculation.
So, I recommend using a graphing calculator or online graphing tool to plot the first derivative f'(x) and find the approximate values of x where the derivative is close to zero. This will give you the critical points.
Step 2: Determining the nature of the critical points.
Once we have the critical points, we need to analyze the behavior of the function around these points to determine whether they are local maxima or minima.
To do this, we can examine the sign of the first derivative on either side of each critical point.
- If the first derivative changes from negative to positive at a critical point, it indicates a local minimum.
- If the first derivative changes from positive to negative at a critical point, it indicates a local maximum.
Step 3: Inflection points.
To find the inflection points, we need to find where the concavity of the function changes. In other words, we need to find where the second derivative equals zero.
The second derivative of the original function f(x) is:
f''(x) = (28/9)(x^(1/3)) + (350/9)(x^(-5/3))
Similar to Step 1, we set the second derivative equal to zero and solve for x:
(28/9)(x^(1/3)) + (350/9)(x^(-5/3)) = 0
Again, this equation is not easy to solve algebraically, so we can use numerical methods to find the approximate values of x where the second derivative is close to zero. These values will be the inflection points.
Step 4: Sketch the graph.
Using the information from Steps 1 to 3, plot the graph of the original function f(x). Mark the critical points and inflection points on the graph. Additionally, determine the behavior (maxima, minima, concavity) of the function based on the analysis of the first and second derivatives.
It is important to remember that the accuracy of your sketch depends on the accuracy of the approximations obtained from numerical methods. Calculators and online tools can provide good estimates, but they might not always be exact.
By following these steps, you should be able to find the local extrema and inflection points of the given function. Remember to double-check your calculations and if needed, use more accurate numerical methods to find the critical points and inflection points.