Question

Find the maximum and minimum values, if any of the following function given by: $f\left(x\right)=9x^2+12x+2$

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 negative then function will take maximum value at that $x$.

So here $f^{{}^{\prime }}\left(x\right)=18x+12$

Putting this equal to $0$ we get
$18x+12=0$
$\Rightarrow x=\frac{-2}{3}$.

Let's look at the second derivative of this function

$f^{{}^{\prime \prime }}\left(x\right)=18$
Which is positive, this means that the given function will take minimum value at $x=\frac{-2}{3}$
Value is given by
$f\left(\frac{-2}{3}\right)=9{\left(\frac{-2}{3}\right)}^2+12\times \frac{-2}{3}+2$
$\Rightarrow f\left(\frac{-2}{3}\right)=-2$