Question

How do you determine of the function $f\left( x \right)$ is concave up and concave down for $f\left( x \right)=2{{x}^{3}}-3{{x}^{2}}-12x+1$?

Answer 1

Hint: Now to find if the function is increasing or decreasing we will find the second derivative of the function. We know that the derivative of the function ${{x}^{n}}$ is given by $n{{x}^{n-1}}$ . Hence using this we will find the second derivative. Now the function is concave up if the second derivative is positive and concave down if the second derivative is negative.

Complete step by step answer:
Now we are given a polynomial function. Now to check if the function is concave up or concave down we check the second derivative of the function. Now whenever the second derivative of the function is positive this means the function is concave up. Similarly when the second derivative is negative the function is concave down.
Now let us consider the function $f\left( x \right)=2{{x}^{3}}-3{{x}^{2}}-12x+1$.
Now we know that the derivative of the function ${{x}^{n}}$ is given by $n{{x}^{n-1}}$ .
Hence using this we can say that,
$f'\left( x \right)=6{{x}^{2}}-6x-12$
Now again differentiating the above function we get,
$f''\left( x \right)=12x-6$
Now we can see that the function obtains the value of 0 at $x=\dfrac{1}{2}$ .
Now for $x>\dfrac{1}{2}$ the second derivative is positive and for $x<\dfrac{1}{2}$ the second derivative is negative.

Hence we get the function is concave up for $x>\dfrac{1}{2}$ and the function is concave down for $x<\dfrac{1}{2}$.

Note: Now note that the second derivative tells us if the function is concave up or concave down. To find if the function is increasing or decreasing we use the first derivative. If the first derivative is greater than 0 then the function is increasing. If the first derivative is less than 0 then the function is decreasing.