Suppose f'(x)

Question

Suppose f'(x) < 2, for all x ≥ 2, and f(2) = 7. Show that f(4) < 11.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the validity of the given statement. If F of X is less than 5, for all X is greater than or equal to 6. And F of 6 is equal to 12, then F of 8 is less than 21. Awesome. So it appears for this particular problem, we're ultimately trying to figure out. If this following statement that's given to us by the problem itself is valid or valid or not, so we're trying to figure out whether or not this statement that's given to us is true or false. That's what we're ultimately trying to solve for. Is this given statement true or false? So with that said, it's either going to be a true or B false. Is this statement true or false? So, now that we know what we're ultimately trying to solve for, Our first step is we need to recall and note that by the mean value theorem, we need to note that there will exist some value of C In parentheses 6,8, such that we could go ahead and write that F of 8 minus F of 6. is going to be equal to F of C. Multiplied by 8 minus 6, which will be all equal to 2 multiplied by F of C, where this prime symbol means we're taking the first derivative of F of C, this function of C. So that's what the prime symbol means. This prime symbol means the first derivative. We also need to recall a note for our second step that since we are given that F of X is less than 5, for all values of X is greater than or equal to 6, that will mean that it falls that F of C is going to be less than 5. So once again, because we know that we're told that F of X is less than 5, for all values of X is greater than or equal to 6, that will mean that we can state that F C is less than 5. Thus, as a result, we can go ahead and write that F of 8 minus F of 6 is less than 2 multiplied by 5, meaning F of 8 minus F of 6 is less than 10. So now we need to note for our third step, we need to note that we're told, or rather we're provided that F of 6 is equal to 12, so therefore we can go ahead and write that F of 8, you could say that F of 8 minus 12. is less than 10, so that means we could say that F of 8 is less than 12 + 10, so that will mean that F of 8 is less than 22. So therefore, the given statement has to be False, and it's false due to the fact that F of 8 is less than 22, because F of 8 is less than 22, that means that our given statement has to be false. So that means that our final answer, without a doubt has to be multiple choice answer letter B, which is false. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Suppose f'(x) < 2, for all x ≥ 2, and f(2) = 7. Show that f(4) < 11.

Key Concepts

Derivative and its Interpretation

Mean Value Theorem

Inequalities and Function Behavior

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