Question

Find the maximum and minimum values, if any of the following function given by:

$f\left(x\right)={(2x-1)}^2+3$

Answer 1

Maximum or minimum can be seen by using derivatives.

Steps1: First find first derivative of the function

Step2: Put it equal to zero and find $x$ were first derivative is zero

Step3: Now find second derivative

Step4: Put $x$ for which first derivative was zero in equation of second derivative

Step5: If second derivative is greater than zero then function takes minimum value at that $x$ and if second derivative is maximum then function will take maximum value at that $x$.

Here $f^{{}^{\prime }}\left(x\right)=8x-4$

Putting this equal to $0$
$f^{{}^{\prime }}\left(x\right)=0$
$\Rightarrow 8x-4=0$
$\Rightarrow x=\frac{1}{2}$.

Second derivative of this function

$f^{{}^{\prime \prime }}\left(x\right)=8$ which is positive.
Which means given function will take minimum value at $x=\frac{1}{2}$
And that is given by
$f\left(\frac{1}{2}\right)=(2\times \frac{1}{2}-1)+3$
$=3$
Maxima does not exist.