# $f$ is convex and $f(10)$, $f(20)$ given. Find the smallest value of $f(7)$.

Mathematics Asked by jixubi on March 15, 2021

If a convex function exists $$f:mathbb Rto mathbb R$$ 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.

You must have $$f(10) =fleft(frac{10}{13} times 7 + frac{3}{13} times 20right) leq frac{10}{13} times f(7) + frac{3}{13} times f(20)$$

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 $$f(7)=-frac{71}{5}$$.

Answered by TheSilverDoe on March 15, 2021

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].

Answered by Kavi Rama Murthy on March 15, 2021