If $f (x) =$ $\left[ \begin{gathered} {x^2}\,\,\,\,\,\,\,\,\,\,\,\,if\,\,\,\,x \leqslant \,{x_0} \hfill \\
ax + b\,\,\,\,\,if\,\,\,\,x\, > \,{x_0} \hfill \\ \end{gathered} \right.$ derivable $\forall \,x\, \in \,R\,\,$ then the values of $a$ and $b$ are respectively
- A$2x_0 , - $ $x_0^2\,$
- B$- x_0 , 2 $ $x_0^2\,$
- C$- 2x_0 , -$ $x_0^2\,$
- D$2x_0^2\,$ , $- x_0$
Similar Questions
Let $P(x)$ be a polynomial with real coefficients such that $P\left(\sin ^2 x\right)=P\left(\cos ^2 x\right)$ for all $x \in[0, \pi / 2)$. Consider the following statements:
$I.$ $P(x)$ is an even function.
$II.$ $P(x)$ can be expressed as a polynomial in $(2 x-1)^2$
$III.$ $P(x)$ is a polynomial of even degree.
Then,
Let $P(x)$ be a polynomial with real coefficients such that $P\left(\sin ^2 x\right)=P\left(\cos ^2 x\right)$ for all $x \in[0, \pi / 2)$. Consider the following statements:
$I.$ $P(x)$ is an even function.
$II.$ $P(x)$ can be expressed as a polynomial in $(2 x-1)^2$
$III.$ $P(x)$ is a polynomial of even degree.
Then,
- [KVPY 2016]
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If $f(x) = \log \frac{{1 + x}}{{1 - x}}$, then $f(x)$ is
Which of the following is true
Which of the following is true
Let $f(x)=\frac{x+1}{x-1}$ for all $x \neq 1$. Let $f^1(x)=f(x), f^2(x)=f(f(x))$ and generally $f^n(x)=f\left(f^{n-1}(x)\right)$ for $n>1$. Let $P=f^1(2) f^2(3) f^3(4) f^4(5)$ Which of the following is a multiple of $P$ ?
Let $f(x)=\frac{x+1}{x-1}$ for all $x \neq 1$. Let $f^1(x)=f(x), f^2(x)=f(f(x))$ and generally $f^n(x)=f\left(f^{n-1}(x)\right)$ for $n>1$. Let $P=f^1(2) f^2(3) f^3(4) f^4(5)$ Which of the following is a multiple of $P$ ?
- [KVPY 2012]
Which of the following function is surjective but not injective
Which of the following function is surjective but not injective