Question

If for a function $f(x)$ , $f'(a) = 0,f''(a) = 0,f'''(a) > 0$ , then find the nature of the function f(x) at x=a.
A. Minimum
B. Maximum
C. Not an extreme point
D. Extreme point

Answer 1

Hint: We will use the conditions of critical points. Then by using the second and third order derivative we will get the required result.

Complete step by step solution:
It is given that $f'(a) = 0,f''(a) = 0,f'''(a) > 0$.
Now, we equate the first derivative of a function to zero to obtain the critical points.
Then we substitute those critical points into the second derivative of the function, if the second derivative of the function is greater than zero at that critical point then we say that the function has minimum value at that point but if the second derivative of the function is less than zero at that critical point then we say that the function is having a maximum value.
But here the second derivative is also zero, so we can not conclude that the function is minimum or maximum at the point a.

Option ‘C’ is correct

Note: Students sometimes get confused with the given inequality $f'''(a) > 0$ and conclude that f(x) has minimum value but for minimum value the second order derivative must be greater than zero not the third order.
Remember that;
f'(a)=0, which means ‘a’ is the critical point.
f"(a)=0, which means the function has neither minima nor maxima at that point.
Also, f'''(a)>0, which means ‘a’ is not an extreme point of the given function.