Let $f$ be a twice differentiable function on $\mathbb{R}$ such that
$f(0)=1, f(1)=2, f^{\prime}(0)=-1, f^{\prime}(1)=5 \text { and } f^{\prime \prime}(x) \geq 0 \text { for } x \in[0,1] .$

Here, $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ denote the first and second order derivative of $f$ at $x$ respectively. Then

• Correct answer
  A

  $f(x) \geq-x+1$ for $x \in[0,1]$
• Correct answer
  B

  $f(x) \geq \frac{1}{3}$ for $x \in[0,1]$
• Correct answer
  C

  $f^{\prime}\left(x_1\right) \leq f^{\prime}\left(x_2\right)$ if $x_1 \lt x_2$ for $x_1, x_2 \in[0,1]$
• Correct answer
  D

  $f(x) \geq 5 x-3$ for $x \in[0,1]$

Hence, the correct answer is options: 'A', 'B', 'C', 'D'.