QJEEM21Mae536: JEE Main 2021 Mathematics - Class 12 JEEM21Mae536 Solution: If f ( x ) = { ∫ 0 x ( 5 +

Question

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If

f (x) = {\begin{matrix} \int_{0}^{x} (5 + | 1 - t |) d t, & x > 2 \\ 5 x + 1, & x \leq 2 \end{matrix}

, then

Accepted Answer

f(x) = \int\limits_0^1 {(5 + (1 - t))dt + \int\limits_1^x {(5 + (t - 1))dt} }

= \left. {6 - {1 \over 2} + \left( {4t + {{{t^2}} \over 2}} \right)} \right|_1^x

= {{11} \over 2} + 4x + {{{x^2}} \over 2} - 4 - {1 \over 2}

= {{{x^2}} \over 2} + 4x - 1

f({2^ + }) = 2 + 8 + 1 = 11

f(2) = f({2^ - }) = 5 \times 2 + 1 = 11

\Rightarrow

continuous at x = 2

Clearly differentiable at x = 1

Lf' (2) = 5

Rf' (2) = 6

\Rightarrow

Not differentiable at x = 2

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