Sketch the graph of a function having the following features:
f'(x)>0 on (-5,-2) and on (3,oo)
f'(x)<0 on (-oo,-5) and on (-2,3)
f''(x)>0 on (-oo,-3) and on (1,oo)
f''(x)<0 on (-3,1)
Please help!
If we assume a polynomial-like function, then we know that it will look kind of like a quartic.
It is concave up on both ends, and has a bump in the middle where it is concave down.
It has minima at x = -5 and x=3, and a maximum at x = -2
Since f"(x) = (x+3)(x-1) = x^2+2x-3
f'(x) = 1/12 x^4 + 1/3 x^3 - 3/2 x^2 + cx + d
I guess you can figure out what c and d are, but the graph will look kind of like
http://www.wolframalpha.com/input/?i=3x%5E4+%2B+16x%5E3+-+66x%5E2+-+360x+for+-6+%3C%3Dx+%3C%3D+5
To sketch the graph of a function with the given features, follow these steps:
1. Determine the critical points of the function: These are the points where the first derivative of the function changes sign or the second derivative changes sign.
- First, find the critical points of the first derivative:
- f'(x) > 0, from -5 to -2
- f'(x) > 0, from 3 to positive infinity
- f'(x) < 0, from negative infinity to -5
- f'(x) < 0, from -2 to 3
- Next, find the critical points of the second derivative:
- f''(x) > 0, from negative infinity to -3
- f''(x) > 0, from 1 to positive infinity
- f''(x) < 0, from -3 to 1
2. Identify the behavior of the function at each critical point:
- At the critical points where f'(x) changes sign from positive to negative (-5 and -2), the function has local maxima.
- At the critical points where f'(x) changes sign from negative to positive (-5 and -2), the function has local minima.
- At the critical points where f''(x) changes sign from positive to negative (-3 and 1), the function has inflection points.
3. Sketch the overall shape of the function:
- Start with an asymptote if there are any (if not mentioned, assume there are none).
- Mark the critical points on the x-axis and label them according to the behavior identified above.
- Connect the critical points with appropriate curves based on the sign changes of the first and second derivatives.
Based on the given information, the graph might look something like this:
*
/ \
/ \
/ \
* * --- *
/ |
/ |
/ |
/ *
-----+-----+-----+-----+-----+-----+-----+-----+-----
-oo -5 -2 1 3 oo
Note: This is a rough sketch, and the exact shape and position of the graph may vary. It is essential to use additional information or actual function values to refine the graph further.