What will be true about the graph of the function
𝑓
, if
𝑓
′
is always positive, but
𝑓
″
is always negative?

Question

What will be true about the graph of the function 
𝑓
, if 
𝑓
′
 is always positive, but 
𝑓
″
is always negative?

BlackTom AI · Accepted Answer

This question asks what is true about the graph of f given that f' is always positive and f'' is always negative. First, recall a key relationship: a positive derivative f' > 0 means the function is increasing wherever the derivative exists. Second, a negative second derivative f'' < 0 means the graph is concave down (the slope f' is decreasing as x increases). Therefore, for every x in the domain, the graph should be increasing and concave down. Now, looking at the provided answer options: the available option (The graph of f will always be increasing and concave down.) correctly captures both properties derived from f' > 0 and f'' < 0. Because the option set is empty in this item (no additional choices to evaluate), there are no alternative statements to assess for correctness or incorrectness here. However, the core reasoning remains that increasing behavior combined with concavity down aligns with f' > 0 and f'' < 0, respectively. In summary, the monotonic increase of f coupled with concavity downward directly follows from the given derivative signs, reaffirming that the graph should indeed be increasing and concave down.

DIFFERENTIAL CALCULUS (MTH_251_020_W2025) Unit 5 - Function Behavior - PreClass

What will be true about the graph of the function 𝑓 , if 𝑓 ′ is always positive, but 𝑓 ″ is always negative?

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