Question

Which of the following condition is true for a monotonically increasing function f(x), which is differentiable in its domain?

• 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 B

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

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 monotonically increasing, so f(b) $\ge$ 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) =

[It is nothing but that if we have a differentiable curve between (a,b) then we’ll find a point ‘c’ such that the slope of tangent at that point will be equal to the slope of the secant at points { a, f(a) } and { b, f(b) } ]

We can also write the above equation as -

$f^{\prime }\left(c\right)(b-a)=f\left(b\right)-f\left(a\right)$

Or $f^{\prime }\left(c\right)(b-a)\ge 0$ as we know that $f\left(b\right)\ge$ f(a)

Or $f^{\prime }\left(c\right)\ge 0$ 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) $\ge$ 0.