GATE DA 2024 Set 1 — Question 37
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Calculus & Optimization → Continuity & Differentiability → Continuity
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Question
Let be a function. Note: denotes the set of real numbers.Which ONE of the following choices gives the values of that make the function continuous and differentiable?
Correct answer
(A) a = (1)/(4), b = 0, c = 1
Solution
For to be continuous at :
So, (Eq. 1)For to be continuous at :
So, (Eq. 2)From (1) and (2), .
Substituting into (1), we get .For differentiability at :
Left derivative:
Right derivative: . At , this is .
So, . Since , .For differentiability at :
Left derivative: . At , this is .
Right derivative: .
So, . Since , . (Consistent)Finally, find using :
.Thus, .
So, (Eq. 1)For to be continuous at :
So, (Eq. 2)From (1) and (2), .
Substituting into (1), we get .For differentiability at :
Left derivative:
Right derivative: . At , this is .
So, . Since , .For differentiability at :
Left derivative: . At , this is .
Right derivative: .
So, . Since , . (Consistent)Finally, find using :
.Thus, .
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