Question

If $f$ be continuous on $[a,b]$ and differentiable on the open interval $(a,b)$, then which of the following is/are correct?

• A. $f$ is strictly increasing in $[a,b]$, if $f^{\prime }(x)>0$ for each $x\in (a,b)$
• B. $f$ is strictly decreasing in $[a,b]$, if $f^{\prime }(x)<0$ for each $x\in (a,b)$
• C. $f$ is a constant function in $[a,b]$, if $f^{\prime }(x)=0$ for each $x\in (a,b)$
• D. All the above are correct

Answer 1

Answer: (D)

Mean value theorem, which states that, if $f:[a,b]\to R$ be a continuous function on $[a,b]$ and differentiable on $(a,b)$, then there exists somec in $(a,b)$ such that
$f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$
Let $x_1,x_2\in [a,b]$ be such that $x_1<x_2$, then there exists a point $c$ between $x_1$ and $x_2$ such that
$f\left(x_2\right)-f\left(x_1\right)=f^{\prime }(c)\left(x_2-x_1\right)$
i.e., $f\left(x_2\right)-f\left(x_1\right)>0$
i.e., $f\left(x_2\right)>f\left(x_1\right)$
Thus, we have
$x_1<x_2\Rightarrow f\left(x_1\right)<f\left(x_2\right)\text{, for all }{x}_1,x_2\in [a,b]$
Hence, $f$ is strictly increasing function in $[a,b]$.
The proofs of part (b) and (c) are similar.
Hence, option (d) is correct.
Remarks
(i) $f$ is strictly increasing in $(a,b)$ if $f^{\prime }(x)>0$ for each $x\in (a,b)$.
(ii) $f$ is strictly decreasing in $(a,b)$ if $f^{\prime }(x)<0$ for each $x\in (a,b)$.
(iii) If a function is strictly increasing or strictly decreasing in an interval $I$, then it is necessarily increasing or decreasing in I. However, converse need not be true.