QJEEM19Maca4a: JEE Main 2019 Mathematics - Class 11 JEEM19Maca4a Solution: Let f(x) = e x – x and g(x) = x 2 – x, x R. Then the set of all …

Question

Verified Solution • 2025–26 Session moderate

Let f(x) = e^x – x and g(x) = x² – x,

\forall

x

\in

R. Then the set of all x

\in

R, where the function h(x) = (fog) (x) is increasing, is :

A [0, $\infty$ )
B $\left[ { - 1, - {1 \over 2}} \right] \cup \left[ {{1 \over 2},\infty } \right)$
C $\left[ { - {1 \over 2},0} \right] \cup \left[ {1,\infty } \right)$
D $\left[ {0,{1 \over 2}} \right] \cup \left[ {1,\infty } \right)$

Accepted Answer

f(x) = ex – x, g(x) = x² – x

f(g(x)) = e^{(x² - x)} - (x² - x)

If f(g(x)) is increasing function

(f (g(x)))^x =

{e^{\left( {{x^2} - x} \right)}} \times (2x - 1) - 2x + 1

\Rightarrow \mathop {(2x - 1)}\limits_A \mathop {[{e^{\left( {{x^2} - x} \right)}} - 1]}\limits_B

A & B are either both positive or negative

for (f (g(x)))^'

\ge

0,

x \in \left[ {0,{1 \over 2}} \right] \cup \left[ {1,\infty } \right)

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