Question

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

1) If \(f'(c) = 0\), then f has a local maximum or minimum at c.

Short answer

The statement is false as \(f\left( x \right)\)does not have a local maximum and minimum.

Step-by-step solution

Example

We know that\(f\left( x \right) = y\),

\(f\left( x \right) = {x^3}\)

Here we differentiate it with respect to x.

\(f'\left( x \right) = \frac{{dy}}{{dx}}\)

\(f'\left( x \right) = 3{x^2}\)

Here the critical point is 0.

Prove that given question statement true or false

Now again we differentiate it with respect to x.

\(f''\left( x \right) = \frac{{{d^2}y}}{{d{x^2}}}\)

\(f''\left( x \right) = 6x\)

Again, differentiate with respect to x.

\(f'''\left( x \right) = \frac{{{d^3}y}}{{d{x^3}}}\)

\(f'''\left( x \right) = 6\)

So, \(f\left( x \right)\)does not have a local maximum and minimum.

Our question statement is false.