Mixing
Counting total number of local maxima and minima of a function
$begingroup$
Find the total number of local maxima and local minima for the function
$$
f(x) = begin{cases}
(2+x)^{3} &text{if}, -3 lt x le -1 \
(x)^frac{2}{3} &text{if}, -1 lt x lt 2
end{cases}
$$
My attempt : I differentiated the function for the two different intervals and obtained the following:
$$
f'(x) = begin{cases}
3cdot(2+x)^{2} &text{if}, -3 lt x le -1 \
frac{2}{3}cdot (x)^frac{-1}{3} &text{if}, -1 lt x lt 2
end{cases}
$$
How do I obtain the maxima and minima points from here.
Any help will be appreciated.
functions maxima-minima
edited Jun 6 '17 at 14:15
Dando18
4,73241235
asked Jun 6 '17 at 14:11
MathsLearnerMathsLearner
694214
$endgroup$
add a comment |
$begingroup$
Find the total number of local maxima and local minima for the function
$$
f(x) = begin{cases}
(2+x)^{3} &text{if}, -3 lt x le -1 \
(x)^frac{2}{3} &text{if}, -1 lt x lt 2
end{cases}
$$
My attempt : I differentiated the function for the two different intervals and obtained the following:
$$
f'(x) = begin{cases}
3cdot(2+x)^{2} &text{if}, -3 lt x le -1 \
frac{2}{3}cdot (x)^frac{-1}{3} &text{if}, -1 lt x lt 2
end{cases}
$$
How do I obtain the maxima and minima points from here.
Any help will be appreciated.
functions maxima-minima
edited Jun 6 '17 at 14:15
Dando18
4,73241235
asked Jun 6 '17 at 14:11
MathsLearnerMathsLearner
694214
$endgroup$
1
$begingroup$
extrema occur where $f'(x) = 0$, $f'(x)$ is undefined, and/or endpoints.
$endgroup$
– Dando18
Jun 6 '17 at 14:12
$begingroup$
@Dando18, "extrema may occur where..." In the problem given, $f'(-2)=0$ but $(-2,0)$ is not an extrema.
$endgroup$
– Bernard Massé
Jun 6 '17 at 14:28
$begingroup$
@BernardMassé Yeah I meant $ text{extrema} implies f'(x)=0,, dots$ not the other way around.
$endgroup$
– Dando18
Jun 6 '17 at 14:30
$begingroup$
Just draw the graph?!
$endgroup$
– Pieter21
Jun 6 '17 at 15:30
add a comment |
$begingroup$
Find the total number of local maxima and local minima for the function
$$
f(x) = begin{cases}
(2+x)^{3} &text{if}, -3 lt x le -1 \
(x)^frac{2}{3} &text{if}, -1 lt x lt 2
end{cases}
$$
My attempt : I differentiated the function for the two different intervals and obtained the following:
$$
f'(x) = begin{cases}
3cdot(2+x)^{2} &text{if}, -3 lt x le -1 \
frac{2}{3}cdot (x)^frac{-1}{3} &text{if}, -1 lt x lt 2
end{cases}
$$
How do I obtain the maxima and minima points from here.
Any help will be appreciated.
functions maxima-minima
edited Jun 6 '17 at 14:15
Dando18
4,73241235
asked Jun 6 '17 at 14:11
MathsLearnerMathsLearner
694214
$endgroup$
Find the total number of local maxima and local minima for the function
$$
f(x) = begin{cases}
(2+x)^{3} &text{if}, -3 lt x le -1 \
(x)^frac{2}{3} &text{if}, -1 lt x lt 2
end{cases}
$$
My attempt : I differentiated the function for the two different intervals and obtained the following:
$$
f'(x) = begin{cases}
3cdot(2+x)^{2} &text{if}, -3 lt x le -1 \
frac{2}{3}cdot (x)^frac{-1}{3} &text{if}, -1 lt x lt 2
end{cases}
$$
How do I obtain the maxima and minima points from here.
Any help will be appreciated.
functions maxima-minima
functions maxima-minima
edited Jun 6 '17 at 14:15
Dando18
4,73241235
asked Jun 6 '17 at 14:11
MathsLearnerMathsLearner
694214
edited Jun 6 '17 at 14:15
Dando18
4,73241235
asked Jun 6 '17 at 14:11
MathsLearnerMathsLearner
694214
edited Jun 6 '17 at 14:15
Dando18
4,73241235
edited Jun 6 '17 at 14:15
Dando18
4,73241235
edited Jun 6 '17 at 14:15
Dando18
4,73241235
4,73241235
asked Jun 6 '17 at 14:11
MathsLearnerMathsLearner
694214
asked Jun 6 '17 at 14:11
MathsLearnerMathsLearner
694214
asked Jun 6 '17 at 14:11
MathsLearnerMathsLearner
694214
694214
1
$begingroup$
extrema occur where $f'(x) = 0$, $f'(x)$ is undefined, and/or endpoints.
$endgroup$
– Dando18
Jun 6 '17 at 14:12
$begingroup$
@Dando18, "extrema may occur where..." In the problem given, $f'(-2)=0$ but $(-2,0)$ is not an extrema.
$endgroup$
– Bernard Massé
Jun 6 '17 at 14:28
$begingroup$
@BernardMassé Yeah I meant $ text{extrema} implies f'(x)=0,, dots$ not the other way around.
$endgroup$
– Dando18
Jun 6 '17 at 14:30
$begingroup$
Just draw the graph?!
$endgroup$
– Pieter21
Jun 6 '17 at 15:30
add a comment |
1
$begingroup$
extrema occur where $f'(x) = 0$, $f'(x)$ is undefined, and/or endpoints.
$endgroup$
– Dando18
Jun 6 '17 at 14:12
$begingroup$
@Dando18, "extrema may occur where..." In the problem given, $f'(-2)=0$ but $(-2,0)$ is not an extrema.
$endgroup$
– Bernard Massé
Jun 6 '17 at 14:28
$begingroup$
@BernardMassé Yeah I meant $ text{extrema} implies f'(x)=0,, dots$ not the other way around.
$endgroup$
– Dando18
Jun 6 '17 at 14:30
$begingroup$
Just draw the graph?!
$endgroup$
– Pieter21
Jun 6 '17 at 15:30
1
1
$begingroup$
extrema occur where $f'(x) = 0$, $f'(x)$ is undefined, and/or endpoints.
$endgroup$
– Dando18
Jun 6 '17 at 14:12
$begingroup$
extrema occur where $f'(x) = 0$, $f'(x)$ is undefined, and/or endpoints.
$endgroup$
– Dando18
Jun 6 '17 at 14:12
$begingroup$
@Dando18, "extrema may occur where..." In the problem given, $f'(-2)=0$ but $(-2,0)$ is not an extrema.
$endgroup$
– Bernard Massé
Jun 6 '17 at 14:28
$begingroup$
@Dando18, "extrema may occur where..." In the problem given, $f'(-2)=0$ but $(-2,0)$ is not an extrema.
$endgroup$
– Bernard Massé
Jun 6 '17 at 14:28
$begingroup$
@BernardMassé Yeah I meant $ text{extrema} implies f'(x)=0,, dots$ not the other way around.
$endgroup$
– Dando18
Jun 6 '17 at 14:30
$begingroup$
@BernardMassé Yeah I meant $ text{extrema} implies f'(x)=0,, dots$ not the other way around.
$endgroup$
– Dando18
Jun 6 '17 at 14:30
$begingroup$
Just draw the graph?!
$endgroup$
– Pieter21
Jun 6 '17 at 15:30
$begingroup$
Just draw the graph?!
$endgroup$
– Pieter21
Jun 6 '17 at 15:30
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You need to study $f'$ :
What is the sign of $f'(x), xin ]-3 , 2 [$ ?
A local maxima/minima $x_m$ appears when :
- $f'(x_m) = 0$ or $x_m$ is a remarquable point (here $x_m in {-3,-1,2}$)
- $f'$ changes sign before and after $x_m$
If the second condition is not verified, $x_m$ is an inflection point.
answered Jun 6 '17 at 14:45
FurraneFurrane
1,3781516
$endgroup$
$begingroup$
I think x=0 is another remarkable point here, since f'(x) becomes undefined at x=0 .
$endgroup$
– MathsLearner
Jun 6 '17 at 15:30
$begingroup$
You're absolutely right, I didn't do the $f'$ study before posting my answer but indeed as pointed out by Dando18 if $f'$ is undefined then you should check the nature of the point.
$endgroup$
– Furrane
Jun 8 '17 at 2:38
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You need to study $f'$ :
What is the sign of $f'(x), xin ]-3 , 2 [$ ?
A local maxima/minima $x_m$ appears when :
- $f'(x_m) = 0$ or $x_m$ is a remarquable point (here $x_m in {-3,-1,2}$)
- $f'$ changes sign before and after $x_m$
If the second condition is not verified, $x_m$ is an inflection point.
answered Jun 6 '17 at 14:45
FurraneFurrane
1,3781516
$endgroup$
$begingroup$
I think x=0 is another remarkable point here, since f'(x) becomes undefined at x=0 .
$endgroup$
– MathsLearner
Jun 6 '17 at 15:30
$begingroup$
You're absolutely right, I didn't do the $f'$ study before posting my answer but indeed as pointed out by Dando18 if $f'$ is undefined then you should check the nature of the point.
$endgroup$
– Furrane
Jun 8 '17 at 2:38
add a comment |
$begingroup$
You need to study $f'$ :
What is the sign of $f'(x), xin ]-3 , 2 [$ ?
A local maxima/minima $x_m$ appears when :
- $f'(x_m) = 0$ or $x_m$ is a remarquable point (here $x_m in {-3,-1,2}$)
- $f'$ changes sign before and after $x_m$
If the second condition is not verified, $x_m$ is an inflection point.
answered Jun 6 '17 at 14:45
FurraneFurrane
1,3781516
$endgroup$
$begingroup$
I think x=0 is another remarkable point here, since f'(x) becomes undefined at x=0 .
$endgroup$
– MathsLearner
Jun 6 '17 at 15:30
$begingroup$
You're absolutely right, I didn't do the $f'$ study before posting my answer but indeed as pointed out by Dando18 if $f'$ is undefined then you should check the nature of the point.
$endgroup$
– Furrane
Jun 8 '17 at 2:38
add a comment |
$begingroup$
You need to study $f'$ :
What is the sign of $f'(x), xin ]-3 , 2 [$ ?
A local maxima/minima $x_m$ appears when :
- $f'(x_m) = 0$ or $x_m$ is a remarquable point (here $x_m in {-3,-1,2}$)
- $f'$ changes sign before and after $x_m$
If the second condition is not verified, $x_m$ is an inflection point.
answered Jun 6 '17 at 14:45
FurraneFurrane
1,3781516
$endgroup$
You need to study $f'$ :
What is the sign of $f'(x), xin ]-3 , 2 [$ ?
A local maxima/minima $x_m$ appears when :
- $f'(x_m) = 0$ or $x_m$ is a remarquable point (here $x_m in {-3,-1,2}$)
- $f'$ changes sign before and after $x_m$
If the second condition is not verified, $x_m$ is an inflection point.
answered Jun 6 '17 at 14:45
FurraneFurrane
1,3781516
answered Jun 6 '17 at 14:45
FurraneFurrane
1,3781516
answered Jun 6 '17 at 14:45
FurraneFurrane
1,3781516
answered Jun 6 '17 at 14:45
FurraneFurrane
1,3781516
1,3781516
$begingroup$
I think x=0 is another remarkable point here, since f'(x) becomes undefined at x=0 .
$endgroup$
– MathsLearner
Jun 6 '17 at 15:30
$begingroup$
You're absolutely right, I didn't do the $f'$ study before posting my answer but indeed as pointed out by Dando18 if $f'$ is undefined then you should check the nature of the point.
$endgroup$
– Furrane
Jun 8 '17 at 2:38
add a comment |
$begingroup$
I think x=0 is another remarkable point here, since f'(x) becomes undefined at x=0 .
$endgroup$
– MathsLearner
Jun 6 '17 at 15:30
$begingroup$
You're absolutely right, I didn't do the $f'$ study before posting my answer but indeed as pointed out by Dando18 if $f'$ is undefined then you should check the nature of the point.
$endgroup$
– Furrane
Jun 8 '17 at 2:38
$begingroup$
I think x=0 is another remarkable point here, since f'(x) becomes undefined at x=0 .
$endgroup$
– MathsLearner
Jun 6 '17 at 15:30
$begingroup$
I think x=0 is another remarkable point here, since f'(x) becomes undefined at x=0 .
$endgroup$
– MathsLearner
Jun 6 '17 at 15:30
$begingroup$
You're absolutely right, I didn't do the $f'$ study before posting my answer but indeed as pointed out by Dando18 if $f'$ is undefined then you should check the nature of the point.
$endgroup$
– Furrane
Jun 8 '17 at 2:38
$begingroup$
You're absolutely right, I didn't do the $f'$ study before posting my answer but indeed as pointed out by Dando18 if $f'$ is undefined then you should check the nature of the point.
$endgroup$
– Furrane
Jun 8 '17 at 2:38
add a comment |
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1
$begingroup$
extrema occur where $f'(x) = 0$, $f'(x)$ is undefined, and/or endpoints.
$endgroup$
– Dando18
Jun 6 '17 at 14:12
$begingroup$
@Dando18, "extrema may occur where..." In the problem given, $f'(-2)=0$ but $(-2,0)$ is not an extrema.
$endgroup$
– Bernard Massé
Jun 6 '17 at 14:28
$begingroup$
@BernardMassé Yeah I meant $ text{extrema} implies f'(x)=0,, dots$ not the other way around.
$endgroup$
– Dando18
Jun 6 '17 at 14:30
$begingroup$
Just draw the graph?!
$endgroup$
– Pieter21
Jun 6 '17 at 15:30