JEE Mains · Maths · STD 12 - 6. Application of derivatives
Let ' \(a\) ' be a real number such that the function \(f(x)=a x^{2}+6 x-15, x \in R\) is increasing in \(\left(-\infty, \frac{3}{4}\right)\) and decreasing in \(\left(\frac{3}{4}, \infty\right) .\) Then the function \(g(x)=a x^{2}-6 x+15, x \in R\) has a:
- A local minimum at \(x=-\frac{3}{4}\)
- B local maximum at \(x=\frac{3}{4}\)
- C local minimum at \(\mathrm{x}=\frac{3}{4}\)
- D local maximum at \(\mathrm{x}=-\frac{3}{4}\)
Answer & Solution
Correct Answer
(D) local maximum at \(\mathrm{x}=-\frac{3}{4}\)
Step-by-step Solution
Detailed explanation
\(f(x)=a x^{2}+6 x-15\) \(f^{\prime}=2 a x+6=2 a\left(x+\frac{3}{a}\right)\) \(\Rightarrow-\frac{3}{a}=\frac{3}{4} \Rightarrow a=-4\) Now \(g(x)=-4 x^{2}-6 x+15\) \(g^{\prime}(x) =-8 x-6\) \(\quad\quad=-2\{4 x+3\}\)
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
- For some \(\theta \in\left(0, \frac{\pi}{2}\right),\) if the eccentricity of the hyperbola, \(x^{2}-y^{2} \sec ^{2} \theta=10\) is \(\sqrt{5}\) times the eccentricity of the ellipse, \(x^{2} \sec ^{2} \theta+y^{2}=5,\) then the length of the latus rectum of the ellipse isJEE Mains 2020 Hard
- Equation of the line passing through the points of intersection of the parabola \(x^2 = 8y\) and the ellipse \(\frac{{{x^2}}}{3} + {y^2} = 1\) isJEE Mains 2013 Hard
- The value of \(\lim\limits _{x \rightarrow 1} \frac{\left(x^{2}-1\right) \sin ^{2}(\pi x)}{x^{4}-2 x^{3}+2 x-1}\) is equal toJEE Mains 2022 Medium
- The solution of the differential equation \(\frac{d y}{d x}=-\left(\frac{x^2+3 y^2}{3 x^2+y^2}\right), y(1)=0\) isJEE Mains 2023 Hard
- Let \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2 \hat{i}+4 \hat{j}-5 \hat{k}\) and \(\vec{c}=x \hat{i}+2 \hat{j}+3 \hat{k}, x \in \mathbb{R}\). If \(\vec{d}\) is the unit vector in the direction of \(\vec{b}+\vec{c}\) such that \(\vec{a} \cdot \vec{d}=1\), then \((\vec{a} \times \vec{b}) \cdot \vec{c}\) is equal toJEE Mains 2024 Hard
- If the system of equations \(x+y+z=2\) \(2 x+4 y-z=6\) \(3 x+2 y+\lambda z=\mu\) has infinitely many solutions, thenJEE Mains 2020 Medium
More PYQs from JEE Mains
- Let \(f: R \rightarrow R\) be a function defined as \(f(x)=a \sin \left(\frac{\pi[x]}{2}\right)+[2-x], a \in R\), where [t] is the greatest integer less than or equal to \(t\). If \(\lim _{x \rightarrow-1} f(x)\) exists, then the value of \(\int_{0}^{4} f(x) d x\) is equal to.JEE Mains 2022 Hard
- Let \(\vec{a}, \vec{b}, \vec{c}\) be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle \(\theta\), with the vector \(\vec{a}+\vec{b}+\vec{c}\). Then \(36 \cos ^{2} 2 \theta\) is equal to \(.....\)JEE Mains 2021 Hard
- Consider a circle \((x-\alpha)^2+(y-\beta)^2=50\), where \(\alpha, \beta>0\). If the circle touches the line \(y+x=0\) at the point \(P\), whose distance from the origin is \(4 \sqrt{2}\) , then \((\alpha+\beta)^2\) is equal to ................JEE Mains 2024 Medium
- Let \(P(3\cos\alpha, 2\sin\alpha)\), \(\alpha \neq 0\), be a point on the ellipse \(\dfrac{x^2}{9}+\dfrac{y^2}{4}=1\), \(Q\) be a point on the circle \(x^2+y^2-14x-14y+82=0\) and \(R\) be a point on the line \(x+y=5\) such that the centroid of the triangle \(PQR\) is \(\left(2+\cos\alpha, 3+\dfrac{2}{3}\sin\alpha\right)\). Then the sum of the ordinates of all possible points \(R\) is:JEE Mains 2026 Hard
- The total number of functions,\(f:\{1,2,3,4\} \cdot\{1,2,3,4,5,6\}\) such that \(f (1)+ f (2)= f (3)\), is equal to .JEE Mains 2022 Hard
- The number of \(4-\)letter words, with or without meaning, each consisting of \(2\) vowels and \(2\) consonants, which can be formed from the letters of the word \(UNIVERSE\) without repetition is \(.........\).JEE Mains 2023 Medium