Question

Is Rolle's theorem applicable to `f(x)=tanx`, when 0 < x < r??

Answer 1

No, since `f(0)!=f(r)` when `0 < r< pi/2`. But the Mean Value Theorem is applicable when `0 < r < pi/2`.

It says there is a number `c` (depending on `r`) in the interval `(0,r)` with the property that `sec^{2}(c)=\frac{tan(r)-tan(0)}{r-0}=tan(r)/r` (since `sec^{2}(x)=d/dx(tan(x))`), which is equivalent to `rsec^{2}(c)=tan(r)`.

One thing this implies is that, since `sec^{2}(c) = 1/(cos^{2}(c)) >1` when `0 < c < pi/2`, it follows that `tan(r)>r` when `0 < r < pi/2`. In other words, the graph of `y=tan(x)` is above the graph of `y=x` when `0 < x < pi/2`.