The graph of a function 𝑦 equals 𝑓 of 𝑥 is shown. At which point is d𝑦 by d𝑥 negative but d two 𝑦 by d𝑥 squared positive.
From the figure, we can see that there are five points, 𝐴, 𝐵, 𝐶, 𝐷, and 𝐸. And we need to determine at which of these five points these two facts are true. Let’s consider first of all at which points d𝑦 by d𝑥 is negative. Now, d𝑦 by d𝑥 is the first derivative of this function. And we know that the first derivative of a function gives the slope of its graph. We can see by looking at the figure at which points the graph is sloping downwards. Or we can sketch in tangents at each point to help with this.
Firstly, sketching in a tangent at point 𝐴, we can see that it does indeed slope downwards. So, d𝑦 by d𝑥 is negative at point 𝐴. At points 𝐵 and 𝐶 however, the tangents each slope upwards, which tells us that d𝑦 by d𝑥 will be positive at these two points. At point 𝐷, the tangent appears to be horizontal, which means that d𝑦 by d𝑥 will be equal to zero at point 𝐷, not negative. Finally, at point 𝐸, we sketch in the tangent. And we see that it is indeed sloping downwards. So, d𝑦 by d𝑥 is negative at point 𝐸. We’re, therefore, left with only two possibilities, point 𝐴 and point 𝐸.
The second condition is that d two 𝑦 by d𝑥 squared, that’s the second derivative of our function 𝑦 with respect to 𝑥, must also be positive. Now, the sign of the second derivative of a function is linked to the concavity of the curve. We recall that when a curve is concave upward, its first derivative d𝑦 by d𝑥 is increasing. We can see this from the sketch. The slope of the tangents is changing from negative to zero to positive. And so, its value is getting larger. If d𝑦 by d𝑥 is increasing, then this means that its derivative d two 𝑦 by d𝑥 squared must be positive.
The opposite is true when a graph is concave downward. Its first derivative is decreasing. And so, its second derivative will be negative. We also note that when a curve is concave upward, the tangents to the curve lie below the curve itself. So, if we’re looking for where the second derivative is positive, and hence where the graph is concave upward, we can look at which point the tangents lie below the curve.
Remember that in order to fulfil the first condition, we’ve narrowed our options down to points 𝐴 and 𝐸. From the figure, we can see the at point 𝐴, the tangent is indeed below the curve. And therefore, the curve is concave upward at point 𝐴. And so, the second derivative is positive. However, at point 𝐸, the tangent is above the curve. And so, the curve is concave downward at point 𝐸, meaning that the second derivative will be negative. So, by considering the slope of the curve and then its concavity, we found that the only point at which d𝑦 by d𝑥 is negative, but d two 𝑦 by d𝑥 squared is positive is point 𝐴.