Question

MY NOTES ASK YOUR TEACHER EXAMPLES Suppose that ) --5 and for all values of x. How large can f) possibly be? SOLUTION We are given that is differentiable and therefore continuous) everywhere. In particular, we can apply the Mean Value Theorem on the interval [0, 5]. There exists a number such that AS) - RO) - SO RS) - RO) Urte We are given that Fx) for also in particular we know that (c) RS)-5-T ross+ The largest possible value for S) L . Multiplying both sides of this inequality by S, we have shtes

Answer 1

Suppose that f(o)= - 5 and fres 29 for all x. 1 1 We are given that f is differentiable (and therefore continuous) everywhere. In particular, we can apply the mean value theorem on the interval [0, 5]. there exists number C such that f(5)-f(0) = fro (5-0) so £15)= f(0) + 5. f'co fs) = – 5 + 5 fico ( f(0) =-5) a we have given that f'(x) <g for all & so in particular we ta know that frosg (x=c). Multiplying both sides of this inequality by 5, We have 5. f'co <45. so fe5=-5+5. f'(c) ² - 5+ 45 = 40 45) < 40 " The largest possible value for f(5) is 40