tangent lines of a differentiable function
$begingroup$
Suppose that $f$ is a differentiable function over an interval like $I$ and for every $a, quad ain I quad text{if}quad g_a(x)=f^{'}(a)(x-a)+f(a), quad text{then:} quad g_a(x)=f(x), |x-a|<delta_a,$ for some $delta_a$ ,$delta_a ge0$. and $quad g_a(x) neq f(x) quad ,|x-a| gedelta_a .$
((tangent line of $f$ at $x=a$ is in touch with it's graph only over an interval containing $a$))
Prove that $f$ is convex or concave over the interval ($I$).
Definitions:
$f$ is convex over an interval like $I$, if for every $a & b in I,quad$if $quad a<bquad$then; $frac {f(x)-f(a)}{x-a} le frac{f(b)-f(a)}{b-a}, a<x<b $
$f$ is concave over an interval like $I$, if for every $a & b in I,quad$if $quad a<bquad$then; $frac {f(x)-f(a)}{x-a} ge frac{f(b)-f(a)}{b-a}, a<x<b $
calculus derivatives
asked Dec 21 '18 at 5:01
shapoorshapoor
627
$endgroup$
add a comment |
$begingroup$
Suppose that $f$ is a differentiable function over an interval like $I$ and for every $a, quad ain I quad text{if}quad g_a(x)=f^{'}(a)(x-a)+f(a), quad text{then:} quad g_a(x)=f(x), |x-a|<delta_a,$ for some $delta_a$ ,$delta_a ge0$. and $quad g_a(x) neq f(x) quad ,|x-a| gedelta_a .$
((tangent line of $f$ at $x=a$ is in touch with it's graph only over an interval containing $a$))
Prove that $f$ is convex or concave over the interval ($I$).
Definitions:
$f$ is convex over an interval like $I$, if for every $a & b in I,quad$if $quad a<bquad$then; $frac {f(x)-f(a)}{x-a} le frac{f(b)-f(a)}{b-a}, a<x<b $
$f$ is concave over an interval like $I$, if for every $a & b in I,quad$if $quad a<bquad$then; $frac {f(x)-f(a)}{x-a} ge frac{f(b)-f(a)}{b-a}, a<x<b $
calculus derivatives
asked Dec 21 '18 at 5:01
shapoorshapoor
627
$endgroup$
add a comment |
$begingroup$
Suppose that $f$ is a differentiable function over an interval like $I$ and for every $a, quad ain I quad text{if}quad g_a(x)=f^{'}(a)(x-a)+f(a), quad text{then:} quad g_a(x)=f(x), |x-a|<delta_a,$ for some $delta_a$ ,$delta_a ge0$. and $quad g_a(x) neq f(x) quad ,|x-a| gedelta_a .$
((tangent line of $f$ at $x=a$ is in touch with it's graph only over an interval containing $a$))
Prove that $f$ is convex or concave over the interval ($I$).
Definitions:
$f$ is convex over an interval like $I$, if for every $a & b in I,quad$if $quad a<bquad$then; $frac {f(x)-f(a)}{x-a} le frac{f(b)-f(a)}{b-a}, a<x<b $
$f$ is concave over an interval like $I$, if for every $a & b in I,quad$if $quad a<bquad$then; $frac {f(x)-f(a)}{x-a} ge frac{f(b)-f(a)}{b-a}, a<x<b $
calculus derivatives
asked Dec 21 '18 at 5:01
shapoorshapoor
627
$endgroup$
Suppose that $f$ is a differentiable function over an interval like $I$ and for every $a, quad ain I quad text{if}quad g_a(x)=f^{'}(a)(x-a)+f(a), quad text{then:} quad g_a(x)=f(x), |x-a|<delta_a,$ for some $delta_a$ ,$delta_a ge0$. and $quad g_a(x) neq f(x) quad ,|x-a| gedelta_a .$
((tangent line of $f$ at $x=a$ is in touch with it's graph only over an interval containing $a$))
Prove that $f$ is convex or concave over the interval ($I$).
Definitions:
$f$ is convex over an interval like $I$, if for every $a & b in I,quad$if $quad a<bquad$then; $frac {f(x)-f(a)}{x-a} le frac{f(b)-f(a)}{b-a}, a<x<b $
$f$ is concave over an interval like $I$, if for every $a & b in I,quad$if $quad a<bquad$then; $frac {f(x)-f(a)}{x-a} ge frac{f(b)-f(a)}{b-a}, a<x<b $
calculus derivatives
calculus derivatives
asked Dec 21 '18 at 5:01
shapoorshapoor
627
asked Dec 21 '18 at 5:01
shapoorshapoor
627
asked Dec 21 '18 at 5:01
shapoorshapoor
627
asked Dec 21 '18 at 5:01
shapoorshapoor
627
asked Dec 21 '18 at 5:01
shapoorshapoor
627
627
add a comment |
add a comment |
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