What is the greatest value of b for which and real valued function f that satisfies the following properties must also satisfy f(1)<5

Question

The properties listed are:

1) f is infinitely differentiable on the real numbers

2) $f(0) = 1, f'(0) = 1$, and $f''(0) = 2$

3) $|f'''(x)| < b$ for all x in $[0,1]$

This is question 42 from the GRE math subject test 9367.

My idea was to integrate both sides of the inequality in property three, and plug in the info from property 2 for the value of c (post integration). Is this a valid approach? I get the correct answer for b, which is 12, but I don't know if this is mathematically valid to integrate a magnitude, and integrate both sides of an inequality.

Please help! Thank you.

Answer 1

Taylor says $f(1)= f(0)+f'(0)\cdot 1 +f''(0)\cdot 1^2/2 +f'''(c_x)\cdot 1^3/6 = 1+1+1+f'''(c_x)/6 $ for some $c_x\in (0,1).$ Thus $f(1) \le 3 + b/6.$ This is less than $5$ if $b<12.$ If $b\ge 12,$ then the polynomial $p(x)=1+x +x^2/2 +bx^3/6$ satisfies $p(1)\ge 5.$