# What are the extrema of # f(x)=(x^2 -9)^3 +10# on the interval [-1,3]?

##### 2 Answers

#### Answer:

We have a minima at

#### Explanation:

A maxima is a high point to which a function rises and then falls again. As such the slope of the tangent or the value of derivative at that point will be zero.

Further, as the tangents to the left of maxima will be sloping upwards, then flattening and then sloping downwards, slope of the tangent will be continuously decreasing, i.e. the value of second derivative would be negative.

A minima on the other hand is a low point to which a function falls and then rises again. As such the tangent or the value of derivative at minima too will be zero.

But, as the tangents to the left of minima will be sloping downwards, then flattening and then sloping upwards, slope of the tangent will be continuously increasing or the value of second derivative would be positive.

If second derivative is zero we have a point of

However, these maxima and minima may either be universal i.e. maxima or minima for the entire range or may be localized, i.e. maxima or minima in a limited range.

Let us see this with reference to the function described in the question and for this let us first differentiate

Its first derivative is given by

=

This would be zero for

Hence maxima or minima occurs at points

To find whether it is maxima or minima, let us look at second differential which is

at

at

Hence, we have a local minima at

. graph{(x^2-9)^3+10 [-5, 5, -892, 891]}

#### Answer:

The absolute minimum is

#### Explanation:

The question does not specify whether we are to find relative or absolute extrema, so we will find both.

Relative extrema can occur only at critical numbers. Critical numbers are values of

Absolute extrema on a closed interval can occur at critical numbers in the interval or at enpoints of the interval.

Because the function asked about here is continuous on

**Critical numbers and relative extrema.**

For

Clearly,

Solving

For

for

So, by the first derivative test,

The other critical number in the interval is *at* *in the domain*.

There is **not** universal agreement whether to say that

Some require value *on both sides* to be less, others require values in the domain on either side to be less.

**Absolute Extrema**

The situation for absolute extrema on a closed interval

Find critical numbers in the closed interval. Call the

Calculate the values

In this question we calculate

The minimum is

the maximum is