Question

Let a function $f$ defined as $f:[a,b]\to R$ and $f^{\prime }\left(x\right)\ne 0,$ then which of the following(s) is(are) correct

• A. $f\left(x\right)$ has atleast one extremum point.
• B. $f\left(x\right)$ have atleast two extremum points.
• C. $f\left(x\right)$ have atleast one minima and one maxima.
• D. either $f\left(x\right)$ has atleast one minima or one maxima.

Answer 1

Answer: (A), (B), (D)

either $f\left(x\right)$ has atleast one minima or one maxima.

Given : $f:[a,b]\to R$ and $f^{\prime }\left(x\right)\ne 0$
$\Rightarrow$ at boundary points $x=a,b,$ either $f^{\prime }\left(x\right)>0$ or $f^{\prime }\left(x\right)<0$
$i.e.f\left(x\right)$ is either increasing or decreasing at $x=a,b$
At $x=a:$
$\Rightarrow f\left(a\right)<f\left(a^{+}\right)$ OR $f\left(a\right)>f\left(a^{+}\right)$
$\therefore$ at $x=a,$ either a minima or a maxima exists.
Similarly at $x=b:$ $f\left(x\right)$ is either increasing or decreasing
$\Rightarrow f\left(b\right)<f\left(b^{-}\right)$ OR $f\left(b\right)>f\left(b^{-}\right)$
$\therefore$ at $x=b,$ either a minima or a maxima exists.
So, possible combinations at points $x=a,b$ are :
$(\text{minima},\text{minima}),(\text{minima},\text{maxima}),(\text{maxima},\text{minima}),(\text{maxima},\text{maxima})$