Question

Point of inflection of the function $f\left(x\right)=x^3$ is

• A. 0
• B. 1
• C. 2
• D. -1

Answer 1

The correct option is A

0

Point of inflection is a point where function changes from either concavity to convexity or convexity to concavity. In other words we can say that it is a point where f”(x) = 0 and f”(x) has opposite signs in the neighborhood of the point. So, we will find f”(x) and equate it to zero to find the point of inflection.

Let’s double differentiate the function.

$f\left(x\right)=x^3$

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

f”(x) = 6x

f”(x) = 0 => 6x = 0

=> x =0

So, x = 0 could be a point of inflection.

for x > 0, f"(x) > 0

for x < 0, f"(x) < 0

We can say that x = 0 is a point of inflection for $f\left(x\right)=x^3$

Alternate method -

We can also find that by using f”’(x) (differentiating three times)

if f"'(x) is non zero when f"(x) (double differentiation) is zero. we have a point of inflection.

f”’(x) = 6

Since $f{”}^{\prime }\left(x\right)\ne 0$ .So x = 0 is a point of inflection.