Question

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

$f\left(x\right)=-{(x-1)}^2+10$

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 less than then function will take its maximum value at that $x$.

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

Putting this equal to $0$
$-2x+2=0$
$\Rightarrow x=1$.

Now let's look at the second derivative of this function

$f^{{}^{\prime \prime }}\left(x\right)=-2$
is $-2$ which is negative.
Which means given function will take maximum value at $x=1$ and that value can be found out putting $x=1$ in given function
$f\left(1\right)=-{(1-1)}^2+10$
which is equal to $10$