Rolle'S Theorem And The Mvt Worksheet 3.2 With Answers - Math 141e Calculus Maximus, Penn State School Page 6
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6Calculus Maximus
WS 3.2: Rolle’s Thm & MVT
16. The figure at right shows two parts of the graph of a
[
]
( )
−
function
f x that is continuous on
and
10, 4
(
)
−
differentiable on
10, 4
. It so happens that the
[
]
( )
′
−
derivative
f x
is also continuous on
10, 4
.
[
]
−
(a) Explain why f must have at least one zero in
10, 4
.
(b) Explain why f ′ must also have at least one zero in the
[
]
−
interval
10, 4
. What are these zeros called?
(c) Make a possible sketch of the function with
(d) Make a possible sketch of the function with at
[
]
[
]
one zero of f ′ on the interval
least two zeros of f ′ on the interval
−
−
.
.
10, 4
10, 4
(e) Were the conditions of continuity of f and f ′ necessary to do parts (a) through (d)? Explain.
Yes the conditions of continuity were necessary because discontinuity, like a
jump, would invalidate the claims.
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