I think the article that quoted(1) is different to what the link shows now(2). However, both the article quoted and the one in the link are relevant to the question.
Alzheimer's Disease : In a nutshell, AD is the most common form of dementia. Although the exact cause is still under investigation, several genes have been implicated in the development of AD. One of the things you see in the brain with disease progression is beta amyloid plaque deposition which disrupts brain activity.
Brain Activity : While the brain consumes around 20% of our body's energy at a baseline, an increase in brain activity only increases this consumption by around 5%. <PERSON>, the first to record an EEG stated that "mental work, as I explained elsewhere, adds only a small increment to the cortical work which is going on continuously and not only in the waking state".(2)
However, there are a few 'intrinsic' networks or regions that perform functions such as "information processing for interpreting, responding to and predicting environmental demands". One of the areas active in this state of the brain is referred to as the 'default mode network' (DMN). It's thought that DMN is main the target of the amyloid plaques responsible for Alzheimer's Disease.(2)
However, there are other multimodal regions of the brain that perform processing of various types of information. It's now suggested that "it appears that regions vulnerable to Ab deposition do not simply involve the DMN, but rather comprise multimodal brain regions that are highly interconnected, plastic, and capable of rapid ATP generation"(1).
So it's a variation of activity in these particular areas - not all - that are implicated with Alzheimer's Disease. When you are performing a task, activity in these areas diminish.
Why keep the brain intentionally active : The reason for the hypothesis that actively engaging in challenging activity may reduce the risk of AD is that individuals with a high cognitive reserve have been shown to cope with the pathological changes such as beta amyloid deposition better. Cognitive reserve has been linked to a bunch of factors including "education, occupation, socioeconomic status, social networks and lifelong participation in cognitive and physical activity" and reflects "lifelong patterns of behaviors, endogenous factors (including genetics) and exposure to environmental factors"(1).
Moreover, it's suggested that perhaps higher cognitive reserve increases neuronal efficiency which has been shown to reduce plaque deposition.(1).
So <PERSON>'s question and jp89's answer aren't really contradictory.
1. <PERSON>, et al. Lifespan brain activity, β-amyloid, and Alzheimer's disease. Trends in Cognitive Sciences. 882
2011. 15(11):520-526.
2. Raiche. Two views of the brain. Trends in Cognitive Sciences. 2010. 14(4):180-190.
3. <PERSON>, et al. Imaging the default mode network in aging and dementia. Biochimica et Biophysica Acta (BBA) - Molecular Basis of Disease. 2012. 1822(3):431-441.
4. Harrison's Principles of Internal Medicine. 16th <PERSON>. 495