A vertex whose removal in a graph $G$ increases the number of components of $G$ is called a cut vertex. For all $n,c$, we determine the maximum number of connected induced subgraphs in a connected graph with order $n$ and $c$ cut vertices, and also characterise those graphs attaining the bound. Moreover, we show that the cycle has the smallest number of connected induced subgraphs among all cut vertex-free connected graphs. The general case $c>0$ remains an open task. We also characterise the extremal graph structures given both order and number of pendant vertices, and establish the corresponding formulas for the number of connected induced subgraphs. The `minimal' graph in this case is a tree, thus coincides with the structure that was given by Li and Wang~[Further analysis on the total number of subtrees of trees. \emph{Electron. J. Comb.} 19(4), #P48, 2012].