Calculus true or false question

Question

If c is a critical point and f"(c) > 0(That is the second derivative is positive, then f(x) has a maximum at the point c.

Answer 1

take a specific example

y = x^2, an upward opening parabloa with vertex at the origin (0,0) = minimum, global and local

first derivative y'= f'(x) = 2x

f'(0) = 2(0) = 0

2nd derivative = f"(x) = 2 >0

one counterexample makes the statement false

when 2nd derivative >0, the point is a minimum, not a maxmum

Answer 2

Hello,

A critical point is one at which the derivative of the function is 0. This means that the point is either a minimum of a maximum. The second derivative at a point tells you what is the concavity is; concave up (example f(x) = x^2 at 0 or a valley) or concave down (example f(x) = -x^2 at 0 or a peak). Since the second derivative is positive at c, it means that the function is concave up and that the critical point is a minimum. Thus, the statement is false.

Hope this helps!

Ibrahim