Let f(x)=3^{(x^{2}-2)^{3}+4}, x in R. Then wh | vedclass

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

Question

(D) Given $f(x) = 3^{(x^2-2)^3+4} = 81 \cdot 3^{(x^2-2)^3}$.First derivative: $f'(x) = 81 \cdot 3^{(x^2-2)^3} \cdot \ln 3 \cdot 3(x^2-2)^2 \cdot 2x =

Vedclass · Accepted Answer

All,$P, Q$ and $R$

Vedclass · Answer

(D) Given $f(x) = 3^{(x^2-2)^3+4} = 81 \cdot 3^{(x^2-2)^3}$.First derivative: $f'(x) = 81 \cdot 3^{(x^2-2)^3} \cdot \ln 3 \cdot 3(x^2-2)^2 \cdot 2x = (486 \ln 3) \cdot 3^{(x^2-2)^3} \cdot x(x^2-2)^2$.For $P$: $f'(x) = 0$ at $x=0$ and $x=\pm \sqrt{2}$. For $x$ near $0$,$x(x^2-2)^2$ changes sign from negative to positive,so $x=0$ is a point of local minima. Thus,$P$ is true.For $Q$: $f''(x) = 0$ at $x=\sqrt{2}$. Since $f''(x)$ changes sign at $x=\sqrt{2}$ (as $(x^2-2)$ is a factor),$x=\sqrt{2}$ is a point of inflection. Thus,$Q$ is true.For $R$: For $x > \sqrt{2}$,$f'(x) > 0$. Analyzing $f''(x)$,for $x > \sqrt{2}$,$f''(x) > 0$,which implies $f'(x)$ is increasing. Thus,$R$ is true.Therefore,all statements $P, Q,$ and $R$ are true.

Let $f(x)=3^{(x^{2}-2)^{3}+4}, x \in R$. Then which of the following statements are true?
$P: x=0$ is a point of local minima of $f$
$Q: x=\sqrt{2}$ is a point of inflection of $f$
$R: f^{\prime}$ is increasing for $x>\sqrt{2}$

Similar Questions

Let $f(x) = (x - 3)^{2018}(x - 2)^{2019}, x \in R$. If $f(\alpha)$ is a relative maximum of $f$ at $x = \alpha$, then $2\alpha + 3f(\alpha) =$

The vertices of a triangle are $(0, 0)$,$(x, \cos x)$,and $(\sin^3 x, 0)$ where $0 < x < \frac{\pi}{2}$. The maximum area for such a triangle in sq. units is:

The minimum value of the expression $7 - 20x + 11x^2$ is (in $/11$)

If $m$ is the minimum value of $k$ for which the function $f(x) = x\sqrt{kx - x^2}$ is increasing in the interval $[0, 3]$ and $M$ is the maximum value of $f$ in $[0, 3]$ when $k = m$,then the ordered pair $(m, M)$ is equal to

Find the total number of local maxima and local minima of the function $f(x) = \begin{cases} (2+x)^3, & -3 < x \leq -1 \\ x^{2/3}, & -1 < x < 2 \end{cases}$

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Let $f(x)=3^{(x^{2}-2)^{3}+4}, x \in R$. Then which of the following statements are true?$P: x=0$ is a point of local minima of $f$$Q: x=\sqrt{2}$ is a point of inflection of $f$$R: f^{\prime}$ is increasing for $x>\sqrt{2}$