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