Question
Transcribed Text
2. Use the graph of f(x) below to answer the given questions.
(a) Where f
0
(x) = 0 x = −3, x = 2
(b) Where f
0
(x) is undefined. x = −1, x = 5
(c) Find all local maximum and minimums of f(x) local Minimum at x = −3 local Max at x = 5
(d) Find all the absolute maximums and minimums of f(x)
None
3. Using the graph and your answers to the previous question, do the following;
label all the critical points on a number line, determine the sign of the derivative of x between each
critical point. How does this information on the number line help you determine which critical
points correspond to local maximums, minimums, or neither.
4. In each case, sketch a graph of a CONTINUOUS function with the given properties.
(a) f
0
(−1) = 0 and f
0
(3) = 0
(b) g
0
(1) = 0and g
0
(4)is undefined
(c) h
0
(−2) = 0and h
0
(2) = 0 and h
0
(0) is undefined
5. Use calculus to determine the (a) critical points, (b) local extrema, (a) intervals on which the
function is increasing or decreasing.
(a) f(x) = x
4 + 2x
3 − 1
f
0
(x) = 4x
3 + 6x
2
0 = 2x
2
(2x + 3)
x = 0, x = −3/2
f
0
(x)
-3/2 0
− + +
Inc: (−3/2, 0) ∪ (0, ∞)
Dec: (−∞, −3/2) Min (-3/2, -2.6875)
7. For the following functions find the open intervals on which the function is increasing and decreasing. Identify the functions local and absolute extreme values, if any, saying where they occur.
(a) f(x) = e
√
x
f
0
(x) = e
√
x
x
√
x
0 =
e
√
x
x
√
x
x = 0
f(0) = 1 Absolute Min
f
0
(x)
0
DNE +
(b) f(x) = x
2
ln(x)
f
0
(x) = 2x ln(x) + x
0 = x(2 ln(x) + 1)
x = 0, x = e
−1/2
f(0) = DNE, f(e
−1/2 = −.18394)AB and Local Min
f
0
(x)
0 e
−1/2
− +
(c) f(x) = x
2/3
(x + 5) f
0
(x) = x
5/3 + 5x
2/3
0 =
5x + 10
3x
1/3
x = 0, x = −2, f(0) = 0Min, f(−2) ≈ 4.762203 Max
f
0
(x)
−2 0
+ − +
8. Sketch the graph of the differential function y = f(x) through the point (1, 1) if f
0
(1) = 0 and
f
0
(x) > 0 for x > 1 and f
0
(x) < 0 for x < 1
9. Given the graph of f
0 below, find the following. Then sketch the graph of f if f is continuous and
f(−6) = 0
(a) The intervals where f is increasing and decreasing.
(b) The x values of any local extrema
(c) The end behavior of the graph.
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