Observe the statements given below: Assertion (A): f(x)=x e-x has the maximum at x=1 Reason (R): f'(1)=0 and f''(1)

Question

Observe the statements given below: Assertion (A): f(x)=x e-x has the maximum at x=1 Reason (R): f'(1)=0 and f''(1)<0 Which of the following is correct? (A) Both (A) and (R) are true and (R) is the correct reason for (A) (B) Both (A) and (R) are true, but (R) is not the correct reason for (A) (C) (A) is true, (R) is false (D) (A) is false, (R) is true

Tardigrade Admin · Accepted Answer

Given, f(x) =x e-x f'(x) =e-x-x e-x f''(x) =-e-x-e-x+x e-x =-2 e-x+x e-x For maximum, put f'(1)=0 ⇒ x=1 and f''(1)=-1<0 ∴ Both A and R are true and R is the correct reason for A.

Q. Observe the statements given below :
Assertion (A) : $f (x) = x e^{- x}$ has the maximum at $x = 1$
Reason (R) : $f^{'} (1) = 0$ and $f^{''} (1) < 0$
Which of the following is correct?

Both (A) and (R) are true and (R) is the correct reason for (A)

Both (A) and (R) are true, but (R) is not the correct reason for (A)

(A) is true, (R) is false

(A) is false, (R) is true

Given, $f (x) = x e^{- x}$
$f^{'} (x) = e^{- x} - x e^{- x}$
$f^{''} (x) = - e^{- x} - e^{- x} + x e^{- x}$
$= - 2 e^{- x} + x e^{- x}$
For maximum, put
$f^{'} (1) = 0 \Rightarrow x = 1$ and $f^{''} (1) = - 1 < 0$
$∴$ Both $A$ and $R$ are true and $R$ is the correct reason for $A$ .

Q. Observe the statements given below : Assertion (A) : f(x)=xe−x has the maximum at x=1 Reason (R) : f′(1)=0 and f′′(1)<0 Which of the following is correct?