JEE Mains · Maths · STD 12 - 6. Application of derivatives
Let \(f(x)=3^{\left(x^{2}-2\right)^{3}+4}, x \in R\). 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 \(x>\sqrt{2}\)
- A Only \(P\) and \(Q\)
- B Only \(P\) and \(R\)
- C Only \(Q\) and \(R\)
- D All, \(P, Q\) and \(R\)
Answer & Solution
Correct Answer
(D) All, \(P, Q\) and \(R\)
Step-by-step Solution
Detailed explanation
\(f(x)=81.3^{\left(x^{2}-2\right)^{3}}\) \(f^{\prime}(x)=81.3^{\left(x^{2}-2\right)^{3}} \ln 3.3\left(x^{2}-2\right)^{2} \cdot 2 x\) \(=(81 \times 6) 3^{\left(x^{2}-2\right)^{3}} x\left(x^{2}-2\right)^{2} \ln 3\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Let \(A=\{1,6,11,16, \ldots\}\) and \(B=\{9,16,23,30, \ldots\}\) be the sets consisting of the first 2025 terms of two arithmetic progressions. Then \(n(A \cup B)\) isJEE Mains 2025 Easy
- The area enclosed by the curves \(y^2+4 x=4\) and \(y-2 x=2\) is :JEE Mains 2023 Medium
- Let the system of equations :
\(\begin{aligned}
& 2 x+3 y+5 z=9 \\
& 7 x+3 y-2 z=8 \\
& 12 x+3 y-(4+\lambda) z=16-\mu
\end{aligned}\)
have infinitely many solutions. Then the radius of the circle centred at \((\lambda, \mu)\) and touching the line \(4 x=3 y\) isJEE Mains 2025 Medium - Let \(C\) be a circle having centre in the first quadrant and touching the \(x\)-axis at a distance of \(3\) units from the origin. If the circle \(C\) has an intercept of length \(6\sqrt{3}\) on \(y\)-axis, then the length of the chord of the circle \(C\) on the line \(x - y = 3\) is :JEE Mains 2026 Medium
- Let a circle \(C\) have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of \(C\) on the line \(x + y = 1\) is \(\sqrt{14}\), then the square of the radius of \(C\) is _______.JEE Mains 2026 Hard
- The minimum distance between any two points \(P _{1}\) and \(P _{2}\) while considering point \(P _{1}\) on one circle and point \(P _{2}\) on the other circle for the given circles' equations \(x^{2}+y^{2}-10 x-10 y+41=0\) \(x^{2}+y^{2}-24 x-10 y+160=0\) is .........JEE Mains 2021 Hard
More PYQs from JEE Mains
- If the vertices of a hyperbola be at \((-2, 0)\) and \((2, 0)\) and one of its foci be at \((-3, 0)\), then which one of the following points does not lie on this hyperbola?JEE Mains 2019 Hard
- If \(f(x)\, = \,2\,{\tan ^{ - 1}}\,x\, + \,{\sin ^{ - 1}}\,\left( {\frac{{2x}}{{1 + {x^2}}}} \right),x > 1\,\) then \(f\,(5)\) is equal toJEE Mains 2015 Hard
- Let \(f : R \rightarrow R\) be a differentiable function such that \(f \left(\frac{\pi}{4}\right)=\sqrt{2}, f \left(\frac{\pi}{2}\right)=0\) and \(f ^{\prime}\left(\frac{\pi}{2}\right)=1\) and let \(g(x)=\int\limits_{x}^{\pi / 4}\left(f^{\prime}(t) \sec t+\tan t \operatorname{sectf}(t)\right) d t\) for \(x \in\left[\frac{\pi}{4}, \frac{\pi}{2}\right)\). Then \(\lim\limits _{ x \rightarrow\left(\frac{\pi}{2}\right)^{-}} g ( x )\) is equal toJEE Mains 2022 Hard
- If \(A=\left(\begin{array}{cc}\frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{-2}{\sqrt{5}} & \frac{1}{\sqrt{5}}\end{array}\right), B=\left(\begin{array}{ll}1 & 0 \\ i & 1\end{array}\right), i=\sqrt{-1}\), and \(\mathrm{Q}=\mathrm{A}^{\mathrm{T}} \mathrm{BA}\), then the inverse of the matrix \(\mathrm{A} \mathrm{Q}^{2021} \mathrm{~A}^{\mathrm{T}}\) is equal to :JEE Mains 2021 Hard
- If the line, \(2 x-y+3=0\) is at a distance \(\frac{1}{\sqrt{5}}\) and \(\frac{2}{\sqrt{5}}\) from the lines \(4 x-2 y+\alpha=0\) and \(6 x-3 y+\beta=0,\) respectively, then the sum of all possible values of \(\alpha\) and \(\beta\) isJEE Mains 2020 Medium
- The integral \(\int_0^\pi \frac{(x+3) \sin x}{1+3 \cos ^2 x} d x\) is equal to :JEE Mains 2025 Medium