While I was in school, I learned a method of thinking about warming up from a brilliant professor named Rhonda Soikowski. She thought of all aspects of our body to be connected. As such, all needed attention during a warm up in order to support the others. The physical body supports the voice, the mind supports the emotional self, and all four could be used to influence any of the others. This goes beyond breath support from the diaphragm or the speed at which the nervous system can transmit a signal from the brain to the body, although those are indeed connected. There is quite a bit of study to back this connection up as well. All exercise is associated with gains in cognitive function. 
As one gets older, often a concern people have is a loss of memory or mental function. Occasionally someone may try a pill to treat this condition. Those pills don’t work for everyone, and frequently the body builds up a tolerance and the effect wears off over time . There are behaviors that could help reduce the likelihood of developing such a condition, even if there are genetic traits that predispose someone to this kind of mental deterioration. For example, it has been found that inactivity dramatically increased the risk of developing dementia for people without the apolipoprotein E genotype, which is a genetic marker for dementia risk . Most people read while seated, so that continued behavior could be a contributing cause to dementia. Simply walking 4000 steps a day thickens the hippocampus (which is associated with memory and spatial function) and surrounding regions, and led to higher performance in attention and information-processing speed and executive function. 
Habits are easier to maintain than they are to start anew. When one has a style of living, it is easier to continue down the path set in front of oneself than it is to change course, even if the change could be for the better. As such, it is better to begin a healthy style of living while young so that the habit can be maintained. There are also a wide variety of benefits for mental functioning for the young body. Higher levels of aerobic/muscular fitness is correlated with improved working memory. Some manners that this could be physiologically induced include the release of immune cells (cytokines, peptides) that uniquely affect brain metabolism and function, which occurs during muscular contraction inherent with exercise. Other benefits from exercise that have an influence on the brain include insulin resistance, decreased adiposity, and improved metabolic control, which are all associated with an improved working memory. 
Just because one did not start healthy behavior in one’s youth does not mean that one should excuse oneself from changing to be the best that one could be. One can still make changes, and one should do so for oneself, not out of guilt or shame based on an external authorit. It is easier to change a habit if the change is for oneself, as an intrinsic motivation, as opposed to an extrinsic motivation reliant on an external source. Choose a specific time to fulfill the behavior, and do it every time you are in that situation.  Just because someone didn’t make the change while young does not provide an excuse to avoid becoming ones best self. Just take one step at a time, and progress toward your ideal life a little more every day.
One's diet can also influence the mind, as well as the physical action one's mind has induces the body to engage in. As Hippocrates said, let food be thy medicine. The physical body and the brain are tightly intertwined. Some ways to improve brain functioning include things like green tea (l-theanine , epigallocatechin-3-gallate ) inhibit free radicals, and caffeine  helps maintain brain function. Nuts and some oils have oleic acid  and polyphenols  which diminish inflammation and promote nerve growth and improve blood flow to the brain. Walnuts and fish have omega-3 fatty acids  which activate nerve growth and suppress inflammation. Flavonoids from berries enhance memory in older individuals with excess weight . Curcumin, which can be found naturally in turmeric, is a magnificent anti-inflammatory, anti-oxidant, anti-viral, and can reduce the severity of an inflammatory cytokine storm . Vitamin D deficiency is associated with dementia , along with a whole host of other problems. Magnesium supplementation helps treat neurodegenerative disease, such as dementia .
If strenuous activity is not something that your body can take at the moment, Tai Chi has tremendous benefits available for those who wish to practice. This long lasting tradition has been found to support one improving balance and reduce the incidence of falls, at least in the short term . It also seems to reduce stress and depression in adults . Physically, it does seem to improve cardiorespiratory fitness and reduce blood pressure . It also does have some short term impact improving the symptoms of someone suffering from early stage dementia . In this specific field, there is less randomized control study than could properly demonstrate the specific method these improvements are known to occur, but it is very clear that this gentle movement which evolved from ancient practices for self defense is better than simply sitting on the couch night after night.