MHB What is the proof for f(1.1)>-0.1?

[Yankel]

Hello all,

I am not sure how to approach this question:

Let f(x) be a continuous and differentiable function of order 2. Let f''(x) >0 for all values of x. The tangent line to the function at x=1 is y=-x+1. Show that f(1.1)>-0.1.

Thanks!

 

[MarkFL]

I would begin with:

$$f(x)=ax^2+bx+c$$

You are given:

$$a>0$$

$$f(1)=0$$

$$f'(1)=-1$$

Use these data to write $b$ and $c$ in terms of $a$. You should then be able to demonstrate that:

$$f(1.1)=0.01a-0.1>-0.1$$