If \( f(x) \) is given as: \( f(x) = \begin{cases} 3ax - 2b, & x<1 ax + b + 1, & x<1 \end{cases} \) and \( \lim_{x \to 1} f(x) \) exists, then the relation between \( a \) and \( b \) is:
.The function \( f(x) \) is given by: \[ f(x) = \begin{cases} \frac{2}{5 - x}, & x<3 \\ 5 - x, & x \geq 3 \end{cases} \] Which of the following is true
If \[ f(x) = \begin{cases} x^\alpha \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases} \] Which of the following is true?
If the equation of the tangent at (2, 3) on y2 = ax3 + b is y = 4x - 5, then the value of a2 + b2 is:
If Rolle's theorem is applicable for the function \(f(x) = x(x+3)e^{-x/2}\) on \([-3, 0]\), then the value of \(c\) is:
For all x ∈ [0, 2024] assume that f (x) is differentiable. f (0) = −2 and f ′(x) ≥ 5. Then the least possible value of f (2024) is:
Let f(x) = \(\int \frac{x}{(x^2+1)(x^2+3)} dx\). If f(3) = \(\frac{1}{4} \log(\frac{5}{6})\), then f(0) =
\(\int\frac{2\cos 2x}{(1+\sin 2x)(1+\cos 2x)}dx=\)
If \(\lim_{n\rightarrow\infty}[(1+\frac{1}{n^{2}})(1+\frac{4}{n^{2}})(1+\frac{9}{n^{2}})\cdots(1+\frac{n^{2}}{n^{2}})]^{\frac{1}{n}}=ae^{b}\), then a+b=