Question

What is the condition for a strictly increasing differentiable function?

• A. $f^{\prime }\left(x\right)>0$
• B. $f^{\prime }\left(x\right)\ge 0$
• C. $f^{\prime }\left(x\right)<0$
• D. $f^{\prime }\left(x\right)\le 0$

Answer 1

The correct option is A

$f^{\prime }\left(x\right)>0$

We can answer this question in the same way how we answered the condition for monotonically increasing function. Let’s assume we have a function f(x) which is differentiable throughout its domain.

Let’s take two points which are in the domain of f(x). Let them be a & b, such that b > a.

It is given that function is strictly increasing, so f(b) > f(a).

Let’s assume there is a point c, such that c $\in$ (a , b)

Now, with the mean value theorem we can say that

f’(c) =

Or f’(c) (b- a) = f(b) - f(a)

Or f’(c) (b- a) > 0 as we know that f(b) > f(a)

Or f’(c) > 0 as b > a

Here, ‘c’ was some random point between the points a & b which are again random points.

So in a general way we can say that f’(x) > 0.