On the interval [1, e],the greatest value of | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $f(x) = x^2 \log x$.To find the critical points,we find the derivative $f'(x)$:$f'(x) = \frac{d}{dx}(x^2) \cdot \log x + x^2 \cdot \frac{d}{dx

Vedclass · Accepted Answer

$e^2$

Vedclass · Answer

(A) Let $f(x) = x^2 \log x$.To find the critical points,we find the derivative $f'(x)$:$f'(x) = \frac{d}{dx}(x^2) \cdot \log x + x^2 \cdot \frac{d}{dx}(\log x) = 2x \log x + x^2 \cdot \frac{1}{x} = 2x \log x + x = x(2 \log x + 1)$.Setting $f'(x) = 0$,we get $x(2 \log x + 1) = 0$.Since $x \in [1, e]$,$x 
eq 0$. Thus,$2 \log x + 1 = 0 \implies \log x = -\frac{1}{2} \implies x = e^{-1/2} = \frac{1}{\sqrt{e}}$.Since $\frac{1}{\sqrt{e}} \approx 0.606$,which is not in the interval $[1, e]$,there are no critical points in the given interval.Therefore,the maximum value must occur at the endpoints $x = 1$ or $x = e$.$f(1) = 1^2 \cdot \log(1) = 1 \cdot 0 = 0$.$f(e) = e^2 \cdot \log(e) = e^2 \cdot 1 = e^2$.Comparing the values $0$ and $e^2$,the greatest value is $e^2$.

On the interval $[1, e]$,the greatest value of $f(x) = x^2 \log x$ is:

Similar Questions

At what value of $x$ is the function $f(x) = x(1 - x^2)$ maximum for $0 \leq x \leq 2$?

If $f(x) = 2x^3 - 3x^2 - 12x + 5$ and $x \in [-2, 4]$,then the maximum value of the function occurs at which value of $x$?

Let $a, b \in R$ be such that the function $f(x) = \ln|x| + bx^2 + ax, x \neq 0$ has extreme values at $x = -1$ and $x = 2$.
Statement-$1$: $f$ has a local maximum at $x = -1$ and $x = 2$.
Statement-$2$: $a = \frac{1}{2}$ and $b = -\frac{1}{4}$.

If $S_1$ and $S_2$ are respectively the sets of local minimum and local maximum points of the function $f(x) = 9x^4 + 12x^3 - 36x^2 + 25, x \in R$,then

Twenty metres of wire is available to fence off a flower bed in the form of a circular sector. What must the radius of the circle be,if the area of the flower bed is to be the greatest (in $m$)?

Vedclass Test Series

Exam Paper Generator

Online Exam Module