Let {f}( {x}) be a real-valued function such that {f}^{'}( {x}_{0})=0 for some {x}_{0} \in(0,1), and {f}^{' '}( {x})>0 for all {x} \in(0, 1). Then f(x) has

Correct Answer is : exactly one local minimum in (0,1)

two distinct local minima in (0,1)

exactly one local minimum in (0,1)

GATE Electrical Engineering (EE) 2021 Solved Papers

Question : 18 of 65

Marks: +1, -0

Let

{f}( {x})

be a real-valued function such that

{f}^{'}( {x}_{0})=0

for some

{x}_{0} ∈(0,1)

, and

{f}^{' '}( {x})>0

for all

{x} ∈(0

, 1). Then

f(x)

has

Solution:

(1).

{f}( {x})

is real valued function.
(2).

f^{'}(x_{0})=0 ; x_{0} ∈(0,1)

⇒ {x}_{0}

is the stationary point.
(3)

f^{' '}(x)>0 ; ∀ x ∈(0,1)

⇒ f^{' '}(x_{0})>0

∴ {f}( {x})

is minimum at

{x}= {x}_{0}

∴

exactly one local minimum occurs in

(0,1)

and if occurs at '

{x}_{0}

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