JEE Advanced 2021 Mathematics - Class 12 JEEAdv21Ma8fe0 Solution: Which of the following statements is TRUE?

For option (a)

{\psi _1}(x) = {e^{ - x}} + x

\therefore

\psi {'_1}(x) = 1 - {e^{ - x}} \Rightarrow \psi {'_1}(x) = 0

Here,

{\psi _1}(x)

is always increasing.

Now,

{\psi _1}(x) &gt; 1

(

\because

{\psi _1}(0) = 1

)

Thus, option (a) is incorrect.

For option (b)

{\psi _2}(x) = {x^2} - 2x - 2{e^{ - x}} + 2

\Rightarrow \psi {'_2}(x) = 2x - 2 + 2{e^{ - x}}

\Rightarrow \psi '{'_2}(x) = 2 - 2{e^{ - x}} \ge 0,\forall x &gt; 0

\Rightarrow \psi {'_2}(x)

is increasing

\Rightarrow \psi {'_2}(x) &gt; \psi {'_2}(0)

\Rightarrow \psi {'_2}(x) &gt; 0,\forall x &gt; 0

Thus, option (b) is incorrect.

For option (c)

f(x) = 2\int_0^x {(t - {t^2}){e^{ - {t^2}}}dt = ( - {e^{ - {t^2}}})_0^x - 2\int_0^x {{t^2}.{e^{ - {t^2}}}} dt}

= 1 - {e^{ - {x^2}}} - 2\int_0^x {{t^2}\left( {1 - {t^2} + {{{t^4}} \over {2!}} + ...} \right)dt}

= 1 - {e^{ - {x^2}}} - {{2{x^3}} \over 3} + {{2{x^5}} \over 5} - {{2{x^7}} \over {2.7}} + {{2{x^9}} \over {6.9}}...

\Rightarrow f(x) - 1 + {e^{ - {x^2}}} + {{2{x^3}} \over 3} - {{2{x^5}} \over 5} &lt; 0

\left( {0,{1 \over 2}} \right)

Hence, option (c) is also incorrect.

For option (d)

g(x) = \int_0^{{x^2}} {\sqrt t .{e^{ - t}}dt}

Put

t = {u^2} \Rightarrow dt = 2u\,du

\therefore

g(x) = \int_x^x {2.{u^2}{e^{ - {u^2}}}du = \int_0^x {2{u^2}\left( {1 - {u^2} + {{{u^4}} \over {2!}}....} \right)du} }

g(x) = {{2{x^3}} \over 3} - {{2{x^5}} \over 5} + {{2{x^7}} \over {2.7}} - {{2{x^9}} \over {9.6}}...

\Rightarrow g(x) - {{2{x^3}} \over 3} + {{2{x^5}} \over 5} - {{2{x^7}} \over {2.7}} \le 0,\forall x \in \left( {0,{1 \over 2}} \right)

\Rightarrow g(x) \le {{2{x^3}} \over 3} - {{2{x^5}} \over 5} + {1 \over 7}{x^7},\forall x \in \left( {0,{1 \over 2}} \right)

Hence, option (d) is correct.

QJEEAdv21Ma8fe0: JEE Advanced 2021 Mathematics - Class 12 JEEAdv21Ma8fe0 Solution: Which of the following statements is TRUE?