MHB Maxima, minima, and the mvt application

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To determine the interval where the function is definitely increasing, given that f''(x) ≥ -1 and f'(1) = 3, we can derive that f'(x) ≥ -x + 4. This means that f'(x) remains positive for values of x less than 4. Thus, the function is increasing on the interval (-15, 4). The discussion highlights the importance of understanding the relationship between the second and first derivatives in applying the Mean Value Theorem. Overall, the key takeaway is that the function is guaranteed to be increasing up to x = 4.
T Botha
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Hi there

I'm prepping for a big test tomorrow and I'm really struggling with this question:If f′′(x)≥−1, x belongs to (−15,15), and f′(1)=3, find the interval over which x is definitely increasing.I'm struggling with substitution because I just don't seem to have enough values. Is there a formula that gives an answer? Please let me know.

:( this is making me so sad.
 
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Since f''(x)\ge -1, f'(x)\ge -x+ C. Since f'(1)= 3, we know that C can be as large as 4. If f'(x)\ge -x+ 4 it will be positive for x less than 4.
 

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