Question

If at x = n the derivative is 0
b) f (x) = 0
c) we have a minimum but not a maximum
d) we have a maximum but not a minimum
e) we can have a minimum or a maximum

Answer 1

Given that at x = n, the derivative is 0 (f'(x) = 0)

Now points on the function where derivative of function is zero, are known as critical point. And at these points slope of tangent line becomes zero.

(your Option A is missing)

b) function does not necessarily have zero value at critical point. (For ex: y = x^2, function and it's derivative both has zero value at x = 0) (But y = x^2 - 1, Now function does not have zero value at x = 0, but it's derivative is zero at x = 0)

Now from second derivative test, if at critical points, where f'(x) is zero

f''(x) < 0, then that critical point will have minimum

f''(x) > 0, then that critical point will have maximum

f''(x) = 0, then that point is known as Inflection point.

So option C and D are wrong.

Option E is correct, As we can have minimum or maximum (Also that point can be inflection point)

Let me know if you've any query.