Ineed with this question for my revision!and tell me how to do it also!
to get a grade 9 in a gcse exam, bhavik must have a mean of at least 93%, averaged over three exam papers. his mean score on the first two papers is 89%. is it still possible for him to achieve a top grade overall?

Well that depends if his exam board rounds his percentage up, lol. 

So Bhavik needs 93% to get a level 9.

I'm going to assume that he got 89% on both his papers, (but it could be any two numbers that average to give 89%, e.g. 88% and 90%.)

Assuming that he gets 100% for his last paper (which is very unlikely), the mean value would be (89% + 89% + 100%) ÷ 3 = 92.67%.

92.67% isn't quite 93%, and it's also very unlikely for him to get 100% on his last paper (when you take the results of his last two papers in to consideration). 

So the answer is no, it's not possible for him to achieve a top grade overall. He's going to have to stick with a level 8. 

Answer

1 number

explanation

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

 

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.