Question

Is it true that: If f"(c) > 0, then the slope of

the tangent line to the graph of the function at
x = c is positive.

Answer 1

yes

Explanation:

the FIRST derivative of a function tells us the slope of a tangent line to the curve at any point. if is positive, then the curve must be increasing. If is negative, then the curve must be decreasing.

the SECOND derivative gives us the slope of the slope function (in other words how fast the slope of the original function changes, and if it is accelerating up - positive - or if it is avengers down - negative).

so, the first derivative would be fully sufficient to get the answer of if the slope of the function at that point is positive or negative.

but because it is only a "if" condition and not a "if and only if" condition, the statement is still true.

there are enough cases, where the slope is positive, but the second derivative is not > 0 (usually = 0).

but if even the second derivative is positive, then, yes, the slope of the original function must be positive too.