Question

Find the local maxima and local minima for the given function and also find the local maximum and local minimum values $f\left(x\right)=x^3-6x^2+9x+15$

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 $x$. If Second derivative is zero them it means that this is the point of inflection.

$f^{{}^{\prime }}\left(x\right)=3x^2-12x+9$

Putting this equal to zero, we get
$f^{{}^{\prime }}\left(x\right)=0$
$3x^2-12x+9=0$
$\Rightarrow (x-1)(x-3)=0$
$\Rightarrow x=1,3$

Now let's see the double derivative of this function.

$f^{{}^{\prime \prime }}\left(x\right)=6x-12$
At $x=1$
$f^{{}^{\prime \prime }}\left(1\right)=6\times 1-12=-6$
So function will take maximum value at $x=1$, which is given by
$f\left(1\right)=19$
At $x=3$
$f^{{}^{\prime \prime }}\left(3\right)=6\times 3-12=6$
This is positive at $x=3$, so function will take a minimum value at $x=3$.

Minimum value is given by $f\left(3\right)=3^3-6\times 3^2+9\times 3+15=15$

Minimum value of the function is $15$

Maximum value of the function is $19$