Depends on what you mean by smart. There's many form of intelligence, and they're differently valued.
An underrated one that's very apparent and essential for example is emotional intelligence.
Different facet of 'intelligence' are displayed differently, and not the same way from person to person.
To me, a real smart person, they only really say what they need to say to influence a situation. They don't just talk a lot just to be perceived as 'smart', that's someone who's insecure and the type to say "I'm the smartest in the room." without irony.
By the way you ask your question, the simple answer is you can't. It's like Sherlock Holmes, if you really think about it, the conclusion he arrives to are... nonsense, it's mostly luck. But audience can suspend their disbelief enough if it's at least some convincing to it.
The reality is, if you have a team of detectives, or a team of forensics, it's a group effort, they're a team. And if no one notices anything, it's quite rare for someone to 'notice' something unnoticeable, and for that to happen case after case after case. No, that's unrealistic. But again, fiction, Sherlock Holmes, you can still write it like that, and it's still fun enough.
The long answer is, you can, and you need to set things up properly for your character to arrive at said conclusion. But the issue is, you have to also think about, why other characters didn't arrive at said conclusion.
Slumdog Millionaire, for example, does this in a wonderful way, as each answer the character knew and each was represented with a flashback. It was all luck in a way.
Now back to intelligence. Look. The point of language is to communicate. To communicate well, you need to know your audience. E.g. You don't explain something to a 5 years old the same way you would explain it to a PHD student. There's fun videos on Youtube regarding "Different complexity of explanation to 5 years old to 30 years old."
Similarly, if a character is just using big words, they are either 1. They do think like that, and they can't regulate themselves well enough in conversation to moderate their word choice for said conversation to have clarity. Or more often, 2. they are trying to sound smarter than they are. In real life, the latter is more common. As for the former, any smart person would realise communicating by using big words that are lost on the person being spoke to is nonsense, it communicates 1 thing, "I know more words, I'm smarter, haha."
I don't know what you mean by "meant to be one of the smart one." Or how you want your character to be perceived regarding 'smart'. If you mean by solving cases and arriving at the correct conclusion, you have to be tangible. Knife + Blood = Murder.
If you mean, smart as in like, being able to understand other characters well, communicate with them well, work with them well, being able to put others first well, etc. Then you show it through character actions and thoughts. Thoughts minding how others character want to be treated.
Different audience will perceive intelligent in diverse ways. Me, for example. I think it is extremely wise, to know one's limit. No one can be an expert at everything, and it's wise to place certain trust in said experts when the topic at hand is beyond one's one expertise. That's a quality of a 'smart' person. Etc.
"The more you know, the more you can understand others, but the less they can understand you. It's simultaneously lonely and wholesome."