Question

The absolute maximum & minimum values of functions can be found by their monotonic & asymptotic behavior provided they exist. We may agree that finite limiting values may be regarded as absolute maximum or minimum. For example, the absolute maximum value of $\frac{1}{1+x^{2m}}\left(mϵN\right)$ is 1. When $x=0$, on the other side absolute minimum value of the some function is 0, which is limiting value of the function when $x\to -\infty$ or $x\to +\infty$. Sometime $f^{\prime }\left(x\right)=0$ & $f^{\prime \prime }\left(x\right)=0$ for $x=a$ but $f^{{}^{\prime }"}\left(x\right)\ne 0$ for $x=a$, then f(x) is neither absolute maximum nor absolute minimum at $x=a$, then $x=a$ is called point of inflexion.

On the basis of above information answer the following questions.

The function $x^4-4x+1$ will have

• A. Absolute maximum value
• B. Absolute minimum value
• C. Both absolute maximum & minimum values
• D. None of these

Answer 1

The correct option is B Absolute minimum value

Given $f\left(x\right)=x^4-4x+1$
$\Rightarrow$ $f^{\prime }\left(x\right)=4(x^3-1)$, so $f^{\prime \prime }\left(x\right)=12x^2>0\left(\forall xϵR\right)$
$\therefore$ $f^{\prime }\left(x\right)=0$
$\Rightarrow$ $x=1$
& $x^2+x+1=0$ which have no real value of $x$ as $D=b^2-4ac<0$
$\therefore$ $x=1$ is only one extreme point at which f(x) attains absolute minimum value.
Ans: B