Chapter 2.1, Problem 95E

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11th Edition

Ron Larson + 1 other

ISBN: 9781337275361

Textbook Problem

True or False? In Exercises 93-96, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If a function has derivatives from both the right and the left at a point, then it is differentiable at that point.

To determine

**To calculate:** Whether the given statement “If a function has derivatives from both the right and left at a point, then it is differentiable at that point” is true or false.

Explanation

**Given:** If a function has derivatives from both the right and left at a point, then it is differentiable at that point.

It is not necessary for a function to be differentiable if it has derivatives from both left-and right-hand sides of a point. In fact, for a function to be differentiable, the function should not only have left- and right-hand derivatives, but they should also be equal.

One such example is that of the function

Sect-2.1 P-1ESect-2.1 P-2ESect-2.1 P-3ESect-2.1 P-4ESect-2.1 P-5ESect-2.1 P-6ESect-2.1 P-7ESect-2.1 P-8ESect-2.1 P-9ESect-2.1 P-10E

Sect-2.1 P-11ESect-2.1 P-12ESect-2.1 P-13ESect-2.1 P-14ESect-2.1 P-15ESect-2.1 P-16ESect-2.1 P-17ESect-2.1 P-18ESect-2.1 P-19ESect-2.1 P-20ESect-2.1 P-21ESect-2.1 P-22ESect-2.1 P-23ESect-2.1 P-24ESect-2.1 P-25ESect-2.1 P-26ESect-2.1 P-27ESect-2.1 P-28ESect-2.1 P-29ESect-2.1 P-30ESect-2.1 P-31ESect-2.1 P-32ESect-2.1 P-33ESect-2.1 P-34ESect-2.1 P-35ESect-2.1 P-36ESect-2.1 P-37ESect-2.1 P-38ESect-2.1 P-39ESect-2.1 P-40ESect-2.1 P-41ESect-2.1 P-42ESect-2.1 P-43ESect-2.1 P-44ESect-2.1 P-45ESect-2.1 P-46ESect-2.1 P-47ESect-2.1 P-48ESect-2.1 P-49ESect-2.1 P-50ESect-2.1 P-51ESect-2.1 P-52ESect-2.1 P-53ESect-2.1 P-54ESect-2.1 P-55ESect-2.1 P-56ESect-2.1 P-57ESect-2.1 P-58ESect-2.1 P-59ESect-2.1 P-60ESect-2.1 P-61ESect-2.1 P-62ESect-2.1 P-63ESect-2.1 P-64ESect-2.1 P-65ESect-2.1 P-66ESect-2.1 P-67ESect-2.1 P-68ESect-2.1 P-69ESect-2.1 P-70ESect-2.1 P-71ESect-2.1 P-72ESect-2.1 P-73ESect-2.1 P-74ESect-2.1 P-75ESect-2.1 P-76ESect-2.1 P-77ESect-2.1 P-78ESect-2.1 P-79ESect-2.1 P-80ESect-2.1 P-81ESect-2.1 P-82ESect-2.1 P-83ESect-2.1 P-84ESect-2.1 P-85ESect-2.1 P-86ESect-2.1 P-87ESect-2.1 P-88ESect-2.1 P-89ESect-2.1 P-90ESect-2.1 P-91ESect-2.1 P-92ESect-2.1 P-93ESect-2.1 P-94ESect-2.1 P-95ESect-2.1 P-96ESect-2.1 P-97ESect-2.1 P-98ESect-2.2 P-1ESect-2.2 P-2ESect-2.2 P-3ESect-2.2 P-4ESect-2.2 P-5ESect-2.2 P-6ESect-2.2 P-7ESect-2.2 P-8ESect-2.2 P-9ESect-2.2 P-10ESect-2.2 P-11ESect-2.2 P-12ESect-2.2 P-13ESect-2.2 P-14ESect-2.2 P-15ESect-2.2 P-16ESect-2.2 P-17ESect-2.2 P-18ESect-2.2 P-19ESect-2.2 P-20ESect-2.2 P-21ESect-2.2 P-22ESect-2.2 P-23ESect-2.2 P-24ESect-2.2 P-25ESect-2.2 P-26ESect-2.2 P-27ESect-2.2 P-28ESect-2.2 P-29ESect-2.2 P-30ESect-2.2 P-31ESect-2.2 P-32ESect-2.2 P-33ESect-2.2 P-34ESect-2.2 P-35ESect-2.2 P-36ESect-2.2 P-37ESect-2.2 P-38ESect-2.2 P-39ESect-2.2 P-40ESect-2.2 P-41ESect-2.2 P-42ESect-2.2 P-43ESect-2.2 P-44ESect-2.2 P-45ESect-2.2 P-46ESect-2.2 P-47ESect-2.2 P-48ESect-2.2 P-49ESect-2.2 P-50ESect-2.2 P-51ESect-2.2 P-52ESect-2.2 P-53ESect-2.2 P-54ESect-2.2 P-55ESect-2.2 P-56ESect-2.2 P-57ESect-2.2 P-58ESect-2.2 P-59ESect-2.2 P-60ESect-2.2 P-61ESect-2.2 P-62ESect-2.2 P-63ESect-2.2 P-64ESect-2.2 P-65ESect-2.2 P-66ESect-2.2 P-67ESect-2.2 P-68ESect-2.2 P-69ESect-2.2 P-70ESect-2.2 P-71ESect-2.2 P-72ESect-2.2 P-73ESect-2.2 P-74ESect-2.2 P-75ESect-2.2 P-76ESect-2.2 P-77ESect-2.2 P-78ESect-2.2 P-79ESect-2.2 P-80ESect-2.2 P-81ESect-2.2 P-82ESect-2.2 P-83ESect-2.2 P-84ESect-2.2 P-85ESect-2.2 P-86ESect-2.2 P-87ESect-2.2 P-88ESect-2.2 P-89ESect-2.2 P-90ESect-2.2 P-91ESect-2.2 P-92ESect-2.2 P-93ESect-2.2 P-94ESect-2.2 P-95ESect-2.2 P-96ESect-2.2 P-97ESect-2.2 P-98ESect-2.2 P-99ESect-2.2 P-100ESect-2.2 P-101ESect-2.2 P-102ESect-2.2 P-103ESect-2.2 P-104ESect-2.2 P-105ESect-2.2 P-106ESect-2.2 P-107ESect-2.2 P-108ESect-2.2 P-109ESect-2.2 P-110ESect-2.2 P-111ESect-2.2 P-112ESect-2.2 P-113ESect-2.2 P-114ESect-2.2 P-115ESect-2.3 P-1ESect-2.3 P-2ESect-2.3 P-3ESect-2.3 P-4ESect-2.3 P-5ESect-2.3 P-6ESect-2.3 P-7ESect-2.3 P-8ESect-2.3 P-9ESect-2.3 P-10ESect-2.3 P-11ESect-2.3 P-12ESect-2.3 P-13ESect-2.3 P-14ESect-2.3 P-15ESect-2.3 P-16ESect-2.3 P-17ESect-2.3 P-18ESect-2.3 P-19ESect-2.3 P-20ESect-2.3 P-21ESect-2.3 P-22ESect-2.3 P-23ESect-2.3 P-24ESect-2.3 P-25ESect-2.3 P-26ESect-2.3 P-27ESect-2.3 P-28ESect-2.3 P-29ESect-2.3 P-30ESect-2.3 P-31ESect-2.3 P-32ESect-2.3 P-33ESect-2.3 P-34ESect-2.3 P-35ESect-2.3 P-36ESect-2.3 P-37ESect-2.3 P-38ESect-2.3 P-39ESect-2.3 P-40ESect-2.3 P-41ESect-2.3 P-42ESect-2.3 P-43ESect-2.3 P-44ESect-2.3 P-45ESect-2.3 P-46ESect-2.3 P-47ESect-2.3 P-48ESect-2.3 P-49ESect-2.3 P-50ESect-2.3 P-51ESect-2.3 P-52ESect-2.3 P-53ESect-2.3 P-54ESect-2.3 P-55ESect-2.3 P-56ESect-2.3 P-57ESect-2.3 P-58ESect-2.3 P-59ESect-2.3 P-60ESect-2.3 P-61ESect-2.3 P-62ESect-2.3 P-63ESect-2.3 P-64ESect-2.3 P-65ESect-2.3 P-66ESect-2.3 P-67ESect-2.3 P-68ESect-2.3 P-69ESect-2.3 P-70ESect-2.3 P-71ESect-2.3 P-72ESect-2.3 P-73ESect-2.3 P-74ESect-2.3 P-75ESect-2.3 P-76ESect-2.3 P-77ESect-2.3 P-78ESect-2.3 P-79ESect-2.3 P-80ESect-2.3 P-81ESect-2.3 P-82ESect-2.3 P-83ESect-2.3 P-84ESect-2.3 P-85ESect-2.3 P-86ESect-2.3 P-87ESect-2.3 P-88ESect-2.3 P-89ESect-2.3 P-90ESect-2.3 P-91ESect-2.3 P-92ESect-2.3 P-93ESect-2.3 P-94ESect-2.3 P-95ESect-2.3 P-96ESect-2.3 P-97ESect-2.3 P-98ESect-2.3 P-99ESect-2.3 P-100ESect-2.3 P-101ESect-2.3 P-102ESect-2.3 P-103ESect-2.3 P-104ESect-2.3 P-105ESect-2.3 P-106ESect-2.3 P-107ESect-2.3 P-108ESect-2.3 P-109ESect-2.3 P-110ESect-2.3 P-111ESect-2.3 P-112ESect-2.3 P-113ESect-2.3 P-114ESect-2.3 P-115ESect-2.3 P-116ESect-2.3 P-117ESect-2.3 P-118ESect-2.3 P-119ESect-2.3 P-120ESect-2.3 P-121ESect-2.3 P-122ESect-2.3 P-123ESect-2.3 P-124ESect-2.3 P-125ESect-2.3 P-126ESect-2.3 P-127ESect-2.3 P-128ESect-2.3 P-129ESect-2.3 P-130ESect-2.3 P-131ESect-2.3 P-132ESect-2.3 P-133ESect-2.3 P-134ESect-2.3 P-135ESect-2.3 P-136ESect-2.3 P-137ESect-2.3 P-138ESect-2.4 P-1ESect-2.4 P-2ESect-2.4 P-3ESect-2.4 P-4ESect-2.4 P-5ESect-2.4 P-6ESect-2.4 P-7ESect-2.4 P-8ESect-2.4 P-9ESect-2.4 P-10ESect-2.4 P-11ESect-2.4 P-12ESect-2.4 P-13ESect-2.4 P-14ESect-2.4 P-15ESect-2.4 P-16ESect-2.4 P-17ESect-2.4 P-18ESect-2.4 P-19ESect-2.4 P-20ESect-2.4 P-21ESect-2.4 P-22ESect-2.4 P-23ESect-2.4 P-24ESect-2.4 P-25ESect-2.4 P-26ESect-2.4 P-27ESect-2.4 P-28ESect-2.4 P-29ESect-2.4 P-30ESect-2.4 P-31ESect-2.4 P-32ESect-2.4 P-33ESect-2.4 P-34ESect-2.4 P-35ESect-2.4 P-36ESect-2.4 P-37ESect-2.4 P-38ESect-2.4 P-39ESect-2.4 P-40ESect-2.4 P-41ESect-2.4 P-42ESect-2.4 P-43ESect-2.4 P-44ESect-2.4 P-45ESect-2.4 P-46ESect-2.4 P-47ESect-2.4 P-48ESect-2.4 P-49ESect-2.4 P-50ESect-2.4 P-51ESect-2.4 P-52ESect-2.4 P-53ESect-2.4 P-54ESect-2.4 P-55ESect-2.4 P-56ESect-2.4 P-57ESect-2.4 P-58ESect-2.4 P-59ESect-2.4 P-60ESect-2.4 P-61ESect-2.4 P-62ESect-2.4 P-63ESect-2.4 P-64ESect-2.4 P-65ESect-2.4 P-66ESect-2.4 P-67ESect-2.4 P-68ESect-2.4 P-69ESect-2.4 P-70ESect-2.4 P-71ESect-2.4 P-72ESect-2.4 P-73ESect-2.4 P-74ESect-2.4 P-75ESect-2.4 P-76ESect-2.4 P-77ESect-2.4 P-78ESect-2.4 P-79ESect-2.4 P-80ESect-2.4 P-81ESect-2.4 P-82ESect-2.4 P-83ESect-2.4 P-84ESect-2.4 P-85ESect-2.4 P-86ESect-2.4 P-87ESect-2.4 P-88ESect-2.4 P-89ESect-2.4 P-90ESect-2.4 P-91ESect-2.4 P-92ESect-2.4 P-93ESect-2.4 P-94ESect-2.4 P-95ESect-2.4 P-96ESect-2.4 P-97ESect-2.4 P-98ESect-2.4 P-99ESect-2.4 P-100ESect-2.4 P-101ESect-2.4 P-102ESect-2.4 P-103ESect-2.4 P-104ESect-2.4 P-105ESect-2.4 P-106ESect-2.4 P-107ESect-2.4 P-108ESect-2.4 P-109ESect-2.4 P-110ESect-2.4 P-111ESect-2.4 P-112ESect-2.4 P-113ESect-2.4 P-114ESect-2.4 P-115ESect-2.4 P-116ESect-2.4 P-117ESect-2.4 P-118ESect-2.4 P-119ESect-2.4 P-120ESect-2.4 P-121ESect-2.4 P-122ESect-2.4 P-123ESect-2.4 P-124ESect-2.4 P-125ESect-2.4 P-126ESect-2.5 P-1ESect-2.5 P-2ESect-2.5 P-3ESect-2.5 P-4ESect-2.5 P-5ESect-2.5 P-6ESect-2.5 P-7ESect-2.5 P-8ESect-2.5 P-9ESect-2.5 P-10ESect-2.5 P-11ESect-2.5 P-12ESect-2.5 P-13ESect-2.5 P-14ESect-2.5 P-15ESect-2.5 P-16ESect-2.5 P-17ESect-2.5 P-18ESect-2.5 P-19ESect-2.5 P-20ESect-2.5 P-21ESect-2.5 P-22ESect-2.5 P-23ESect-2.5 P-24ESect-2.5 P-25ESect-2.5 P-26ESect-2.5 P-27ESect-2.5 P-28ESect-2.5 P-29ESect-2.5 P-30ESect-2.5 P-31ESect-2.5 P-32ESect-2.5 P-33ESect-2.5 P-34ESect-2.5 P-35ESect-2.5 P-36ESect-2.5 P-37ESect-2.5 P-38ESect-2.5 P-39ESect-2.5 P-40ESect-2.5 P-41ESect-2.5 P-42ESect-2.5 P-43ESect-2.5 P-44ESect-2.5 P-45ESect-2.5 P-46ESect-2.5 P-47ESect-2.5 P-48ESect-2.5 P-49ESect-2.5 P-50ESect-2.5 P-51ESect-2.5 P-52ESect-2.5 P-53ESect-2.5 P-54ESect-2.5 P-55ESect-2.5 P-56ESect-2.5 P-57ESect-2.5 P-58ESect-2.5 P-59ESect-2.5 P-60ESect-2.5 P-61ESect-2.5 P-62ESect-2.5 P-63ESect-2.5 P-64ESect-2.5 P-65ESect-2.5 P-66ESect-2.5 P-67ESect-2.5 P-68ESect-2.5 P-69ESect-2.5 P-70ESect-2.5 P-71ESect-2.5 P-72ESect-2.5 P-73ESect-2.5 P-74ESect-2.5 P-75ESect-2.5 P-76ESect-2.5 P-77ESect-2.6 P-1ESect-2.6 P-2ESect-2.6 P-3ESect-2.6 P-4ESect-2.6 P-5ESect-2.6 P-6ESect-2.6 P-7ESect-2.6 P-8ESect-2.6 P-9ESect-2.6 P-10ESect-2.6 P-11ESect-2.6 P-12ESect-2.6 P-13ESect-2.6 P-14ESect-2.6 P-15ESect-2.6 P-16ESect-2.6 P-17ESect-2.6 P-18ESect-2.6 P-19ESect-2.6 P-20ESect-2.6 P-21ESect-2.6 P-22ESect-2.6 P-23ESect-2.6 P-24ESect-2.6 P-25ESect-2.6 P-26ESect-2.6 P-27ESect-2.6 P-28ESect-2.6 P-29ESect-2.6 P-30ESect-2.6 P-31ESect-2.6 P-32ESect-2.6 P-33ESect-2.6 P-34ESect-2.6 P-35ESect-2.6 P-36ESect-2.6 P-37ESect-2.6 P-38ESect-2.6 P-39ESect-2.6 P-40ESect-2.6 P-41ESect-2.6 P-42ESect-2.6 P-43ESect-2.6 P-44ESect-2.6 P-45ESect-2.6 P-46ESect-2.6 P-47ESect-2.6 P-48ESect-2.6 P-49ESect-2.6 P-50ECh-2 P-1RECh-2 P-2RECh-2 P-3RECh-2 P-4RECh-2 P-5RECh-2 P-6RECh-2 P-7RECh-2 P-8RECh-2 P-9RECh-2 P-10RECh-2 P-11RECh-2 P-12RECh-2 P-13RECh-2 P-14RECh-2 P-15RECh-2 P-16RECh-2 P-17RECh-2 P-18RECh-2 P-19RECh-2 P-20RECh-2 P-21RECh-2 P-22RECh-2 P-23RECh-2 P-24RECh-2 P-25RECh-2 P-26RECh-2 P-27RECh-2 P-28RECh-2 P-29RECh-2 P-30RECh-2 P-31RECh-2 P-32RECh-2 P-33RECh-2 P-34RECh-2 P-35RECh-2 P-36RECh-2 P-37RECh-2 P-38RECh-2 P-39RECh-2 P-40RECh-2 P-41RECh-2 P-42RECh-2 P-43RECh-2 P-44RECh-2 P-45RECh-2 P-46RECh-2 P-47RECh-2 P-48RECh-2 P-49RECh-2 P-50RECh-2 P-51RECh-2 P-52RECh-2 P-53RECh-2 P-54RECh-2 P-55RECh-2 P-56RECh-2 P-57RECh-2 P-58RECh-2 P-59RECh-2 P-60RECh-2 P-61RECh-2 P-62RECh-2 P-63RECh-2 P-64RECh-2 P-65RECh-2 P-66RECh-2 P-67RECh-2 P-68RECh-2 P-69RECh-2 P-70RECh-2 P-71RECh-2 P-72RECh-2 P-73RECh-2 P-74RECh-2 P-75RECh-2 P-76RECh-2 P-77RECh-2 P-78RECh-2 P-79RECh-2 P-80RECh-2 P-81RECh-2 P-82RECh-2 P-83RECh-2 P-84RECh-2 P-85RECh-2 P-86RECh-2 P-87RECh-2 P-88RECh-2 P-89RECh-2 P-90RECh-2 P-1PSCh-2 P-2PSCh-2 P-3PSCh-2 P-4PSCh-2 P-5PSCh-2 P-6PSCh-2 P-7PSCh-2 P-8PSCh-2 P-9PSCh-2 P-10PSCh-2 P-11PSCh-2 P-12PSCh-2 P-13PSCh-2 P-14PSCh-2 P-15PS

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