The maximum value of the function f(x)=ex+x ln x on the interval 1 ≤ x ≤ 2 is

Question

The maximum value of the function f(x)=ex+x ln x on the interval 1 ≤ x ≤ 2 is (A) e 2+ ln 2+1 (B) e 2+2 ln 2 (C) e π / 2+(π/2) ln (π/2) (D) e 3 / 2+(3/2) ln (3/2)

Tardigrade Admin · Accepted Answer

f'(x)=ex+1+ℓ n x>0 (as x ∈[1,2]) ⇒ f(x) increases in [1,2] ⇒ f max =f(2)= e 2+2 ℓ n 2

Q. The maximum value of the function $f (x) = e^{x} + x ln x$ on the interval $1 \leq x \leq 2$ is

$e^{2} + ln 2 + 1$

$e^{2} + 2 ln 2$

$e^{π /2} + \frac{π}{2} ln \frac{π}{2}$

$e^{3/2} + \frac{3}{2} ln \frac{3}{2}$

$f^{'} (x) = e^{x} + 1 + ℓ n x > 0$
(as $x \in [1, 2]$ )
$\Rightarrow f (x)$ increases in $[1, 2]$
$\Rightarrow f_{m a x} = f (2) = e^{2} + 2 ℓ n 2$

Q. The maximum value of the function f(x)=ex+xlnx on the interval 1≤x≤2 is

e2+ln2+1

e2+2ln2

eπ/2+2π​ln2π​

e3/2+23​ln23​