QJEEM21Maddd2: JEE Main 2021 Mathematics - Class 12 JEEM21Maddd2 Solution: Let f : R R be defined as f ( x ) = { − 4 3 x 3 + 2 x 2 + 3 x , …

Question

Verified Solution • 2025–26 Session moderate

Let f : R

\to

R be defined as

f (x) = {\begin{matrix} - \frac{4}{3} x^{3} + 2 x^{2} + 3 x, & x > 0 \\ 3 x e^{x}, & x \leq 0 \end{matrix}

. Then f is increasing function in the interval

Accepted Answer

f^{'} (x) {\begin{matrix} - 4 x^{2} + 4 x + 3 & x > 0 \\ 3 e^{x} (1 + x) & x \leq 0 \end{matrix}

For x > 0,

f'(x) = - 4{x^2} + 4x + 3

f(x) is increasing in

\left( { - {1 \over 2},{3 \over 2}} \right)

For x

\le

0, f'(x) = 3e^x(1 + x)

f'(x) > 0

\forall

x

\in

(

-

1, 0)

\Rightarrow

f(x) is increasing in (

-

1, 0)

So, in complete domain, f(x) is increasing in

\left( { - 1,{3 \over 2}} \right)

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