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If a convex function exists $f:\mathbb R\to \mathbb R$ and satisfies $f(10) = -4$ and $f(20) = 30$, how should I find the smallest value for $f(7)$?

I have tried finding the linear equations between the two points $f(10)$, $f(20)$ and just input $f(7)$ to get a value since all linear functions are convex/concave. But I am unsure the value I calculated (which is $-71/5$) is the smallest value possible.

Arctic Char
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jixubi
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2 Answers2

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Hint: $10=a(7)+(1-a)(20)$ where $a=\frac {10} {13}$. Hence $-4=f(10) \leq af(7)+(1-a)(30)$. This gives lower bound for $f(7)$ and this value is attained when $f$ is a linear function. [ If you draw the straight line passing through the points $(10,-4), (20,30)$ then you get the graph of a convex function whose value at $7$ is $-\frac {71} 5$ the lower bound you get from above inequality].

Kavi Rama Murthy
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  • What formula should I be referring to understand the comment above? I have solved my question just simply finding the linear function between f(10) and f(20) and substituting f(7) to find the answer – jixubi Aug 18 '20 at 10:34
  • @jixubi I have just use then definition of convexity. Whenever $x<y<z$ we can write $y$ in the form $ax+(1-a)z$ and the use the definition of convexity: $f(y)\leq af(x)+(1-a)f(z)$. – Kavi Rama Murthy Aug 18 '20 at 11:37
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You must have $$f(10) =f\left(\frac{10}{13} \times 7 + \frac{3}{13} \times 20\right) \leq \frac{10}{13} \times f(7) + \frac{3}{13} \times f(20)$$

So $$-4 \leq \frac{10}{13} \times f(7) + \frac{3}{13} \times 30$$

i.e. $$f(7) \geq -\frac{71}{5} $$

Now you can check that it is the lower bound. Indeed, the function $$f : x \mapsto \frac{17}{5}x - 38$$

is convex, satisfies $f(10)=-4$ and $f(20)=30$, and $f(7)=-\frac{71}{5}$.

TheSilverDoe
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  • What formula should I be referring to understand the comment above? I have solved my question just simply finding the linear function between f(10) and f(20) and substituting f(7) to find the answer – jixubi Aug 18 '20 at 10:53
  • First part of my answer shows that for any function with the hypothesis, you must have $f(7) \geq -71/5$. Then I give an explicit example where this bound is attained (same as yours), which proves that the smallest value is exactly $-71/5$. – TheSilverDoe Aug 18 '20 at 11:03
  • Thanks for your reply im still trying to understand how you derive this equation. – jixubi Aug 18 '20 at 11:18
  • Which equation ? – TheSilverDoe Aug 18 '20 at 11:18
  • The first line of your answer :) – jixubi Aug 18 '20 at 11:23
  • This is the definition of convexity :) $f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)$ for all $t \in [0,1]$, and all $x,y \in \mathbb{R}$. – TheSilverDoe Aug 18 '20 at 12:09