Maxima and Minima Exercises

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Here we’ll practice on maxima and minima concepts.

If $f'(a) = 0$ , then $f(a)$ is a local extremum of $f$ .

True False

If $f$ is continuous, $f'(x)<0$ when $x<a$ and $f'(x) < 0$ when $x>a$ , then $f(a)$ is a local max.

True False

$f$ will always have an extremum at $x=a$ , but will it be a maximum?

A function on a closed interval must have a local extremum.

True False

If $f$ has a local max at $x=a$ , then its square $f^2$ must also have a local max at $x=a$

True False

If $f'(a)$ does not exist, then $f$ cannot have a local extremum at $x=a$

True False

If $f$ and $g$ both have local minima at $x=a$ , then their sum $f+g$ has a local minima at $x=a$ as well.

True False

If $f$ is differentiable and decreasing on $(a,b)$ , then $f'(x)<0$ on $(a,b)$

True False

If $f$ and $g$ both have local minima at $x=a$ , then their product $f\cdot g$ has a local minima at $x=a$ as well.

True False