Finding Maxima and Minima

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Math › Finding Maxima and Minima

Questions 1 - 2

1

Consider the function

Find the maximum of the function on the interval $\left [ 0,1 \right ]$ .

Explanation

Notice that on the interval $\left [ 0,1 \right ]$ , the term $\ x^{2}$ is always less than or equal to . So the function is largest at the points when . This occurs at and .

Plugging in either 1 or 0 into the original function yields the correct answer of 0.

2

The function is such that

When you take the second derivative of the function , you obtain

What can you conclude about the function at ?

The point is an inflection point.

The point is a local maximum.

The point is an absolute minimum.

The point is an absolute maximum.

The point is a local minimum.

Explanation

We have a point at which $f^{'}(x)=0$ . We know from the second derivative test that if the second derivative is negative, the function has a maximum at that point. If the second derivative is positive, the function has a minimum at that point. If the second derivative is zero, the function has an inflection point at that point.

Plug in 0 into the second derivative to obtain

So the point is an inflection point.

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