4.3 Maxima and minima  (Page 5/15)

In precalculus, you learned a formula for the position of the maximum or minimum of a quadratic equation $y=a{x}^2+bx+c,$ which was $m=-\frac{b}{\left(2a\right)}.$ Prove this formula using calculus.

If you are finding an absolute minimum over an interval $[a,b],$ why do you need to check the endpoints? Draw a graph that supports your hypothesis.

If you are examining a function over an interval $(a,b),$ for $a$ and $b$ finite, is it possible not to have an absolute maximum or absolute minimum?

When you are checking for critical points, explain why you also need to determine points where $f\left(x\right)$ is undefined. Draw a graph to support your explanation.

Can you have a finite absolute maximum for $y=a{x}^2+bx+c$ over $(\text{−}\infty ,\infty )?$ Explain why or why not using graphical arguments.

Can you have a finite absolute maximum for $y=a{x}^3+b{x}^2+cx+d$ over $(\text{−}\infty ,\infty )$ assuming a is non-zero? Explain why or why not using graphical arguments.

Let $m$ be the number of local minima and $M$ be the number of local maxima. Can you create a function where $M>m+2?$ Draw a graph to support your explanation.

Is it possible to have more than one absolute maximum? Use a graphical argument to prove your hypothesis.

Is it possible to have no absolute minimum or maximum for a function? If so, construct such a function. If not, explain why this is not possible.

[T] Graph the function $y=e^{ax}.$ For which values of $a,$ on any infinite domain, will you have an absolute minimum and absolute maximum?