Let f ( x ) = ( x − 2 ) 2 + 3 for x ≥ 2 , and f ( x ) = k + x 2 for x < 2 . Which are true?

For x < 2 , x 2 already ranges over [ 0 , ∞ ) , so when k ≥ 3 the union with the right branch has range [ 3 , ∞ ) . Continuity at 2 would require k + 4 = 3 , i.e. k = − 1 , but then the one-sided derivatives are 4 and 0 . Thus differentiability fails for every k .

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Mathematics - Application of Derivative Question with Solution | TestHub

MathematicsApplication of DerivativeMiscellaneous/MixedHard2 minQB

MathematicsHardmultiple choice

Let $f (x) = (x - 2)^{2} + 3$ for $x \geq 2$ , and $f (x) = k + x^{2}$ for $x < 2$ . Which are true?

Options:(select one or more)

Answer:

B, C

Solution:

For $x < 2, x^{2}$ already ranges over $[0, \infty)$ , so when $k \geq 3$ the union with the right branch has range $[3, \infty)$ . Continuity at 2 would require $k + 4 = 3$ , i.e. $k = - 1$ , but then the one-sided derivatives are 4 and 0 . Thus differentiability fails for every $k$ .

Stream:JEESubject:MathematicsTopic:Application of DerivativeSubtopic:Miscellaneous/Mixed

⏱ 2mℹ️ Source: QB

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