JEE Mains · Maths · STD 12 - 6. Application of derivatives
The minimum value of the twice differentiable function \(f(x)=\int_{0}^{x} e^{x-t} f^{\prime}(t) d t-\left(x^{2}-x+1\right) e^{x}, x \in R\), is.
- A \(-\frac{2}{\sqrt{ e }}\)
- B \(-2 \sqrt{ e }\)
- C \(-\sqrt{ e }\)
- D \(\frac{2}{\sqrt{ e }}\)
Answer & Solution
Correct Answer
(A) \(-\frac{2}{\sqrt{ e }}\)
Step-by-step Solution
Detailed explanation
\(f(x)=e^{x} \cdot \int_{0}^{x} \frac{f^{\prime}(t)}{e^{t}} d t\) \(f^{\prime}(x)=e^{x} \cdot \int_{0}^{x} \frac{f^{\prime}(t)}{e^{t}} d t+e^{x} \cdot \frac{f^{\prime}(x)}{e^{x}}\) \(-\left[(2 x-1) \cdot e^{x}+\left(x^{2}-x+1\right) \cdot e^{x}\right]\)…
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 \(A_1, A_2, A_3\) be the three A.P. with the same common difference \(d\) and having their first terms as \(A , A +1, A +2\), respectively. Let \(a , b , c\) be the \(7^{\text {th }}, 9^{\text {th }}, 17^{\text {th }}\) terms of \(A_1, A_2, A_3\), respectively such that \(\left|\begin{array}{lll} a & 7 & 1 \\ 2 b & 17 & 1 \\ c & 17 & 1\end{array}\right|+70=0\) If \(a=29\), then the sum of first \(20\) terms of an \(AP\) whose first term is \(c - a - b\) and common difference is \(\frac{ d }{12}\), is equal to \(........\).JEE Mains 2023 Hard
- The sum of all the local minimum values of the twice differentiable function \(\mathrm{F}: \mathrm{R} \rightarrow \mathrm{R}\) defined by \(f(x)=x^{3}-3 x^{2}-\frac{3 f^{\prime \prime}(2)}{2} x+f^{\prime \prime}(1)\) is:JEE Mains 2021 Hard
- The number of seven digits odd numbers, that can be formed using all the seven digits \(1, 2, 2, 2, 3, 3, 5\) is \(.......\)JEE Mains 2023 Hard
- Let L be the line \( \frac{x+1}{2}=\frac{y+1}{3}=\frac{z+3}{6} \) and let S be the set of all points (a, b, c) on L, whose distance from the line \( \frac{x+1}{2}=\frac{y+1}{3}=\frac{z+9}{0} \) along the line L is 7. Then \( \sum_{(a,b,c)\in S}\ (a+b+c) \) is equal to:JEE Mains 2026 Medium
- Let \(x=2 t, y=\frac{i}{3}\) be a conic. Let \(S\) be the focus and \(B\) be the point on the axis of the conic such that \(SA \perp BA\), where \(A\) is any point on the conic. If \(k\) is the ordinate of the centroid of \(\Delta SAB\), then \(\lim _{ t \rightarrow 1} k\) is equal toJEE Mains 2022 Hard
- Suppose that a function \(f: R \rightarrow R\) satisfies \(f(x+y)=f(x) f(y)\) for all \(x, y \in R\) and \(f(1)=3 .\) If \(\sum \limits_{i=1}^{n} f(i)=363,\) then \(n\) is equal toJEE Mains 2020 Medium
More PYQs from JEE Mains
- If for the complex numbers \(z\) satisfying \(|z-2-2 i| \leq 1\), the maximum value of \(|3 i z+6|\) is attained at \(\mathrm{a}+i \mathrm{~b}\), then \(\mathrm{a}+\mathrm{b}\) is equal to .... .JEE Mains 2021 Hard
- Let \(f : R \to R\) be a differentiable function satisfying \(f’’(3) + f’(2) = 0\). Then \(\mathop {\lim }\limits_{x \to \infty } {\left( {\frac{{1 + f\left( {3 + x} \right) - f\left( 3 \right)}}{{1 + f\left( {2 - x} \right) - f\left( 2 \right)}}} \right)^{\frac{1}{x}}}\) is equal toJEE Mains 2019 Hard
- \(\smallint \frac{{{{\sin }^2}x{{\cos }^2}x}}{{{{\left( {{{\sin }^5}x + {{\cos }^3}x{{\sin }^2}x + {{\sin }^3}x{{\cos }^2}x + {{\cos }^5}x} \right)}^2}}}dx\)JEE Mains 2018 Hard
- Let \(S=\mathbf{N} \cup\{0\}\). Define a relation \(R\) from \(S\) to \(\mathbf{R}\) by :
\(\mathrm{R}=\left\{(x, y): \log _{\mathrm{e}} y=x \log _{\mathrm{e}}\left(\frac{2}{5}\right), x \in \mathrm{~S}, y \in \mathbf{R}\right\}\)
Then, the sum of all the elements in the range of \(R\) is equal to :JEE Mains 2025 Medium - If the number of integral terms in the expansion of \(\left(3^{\frac{1}{2}}+5^{\frac{1}{8}}\right)^{\text {n }}\) is exactly \(33,\) then the least value of \(n\) isJEE Mains 2020 Medium
- Let the mirror image of the point \((a, b\), c) with respect to the plane \(3 x-4 y+12 z+19=0\) be (a- \(6, \beta, \gamma)\). If \(a+b+c=5\), then \(7 \beta-9 \gamma\) is equal toJEE Mains 2022 Hard