### Video Transcript

The graph of a function π¦ equals π of π₯ is shown. At which point is dπ¦ by dπ₯ negative but d two π¦ by dπ₯ squared positive.

From the figure, we can see that there are five points, π΄, π΅, πΆ, π·, and πΈ. And we need to determine at which of these five points these two facts are true. Letβs consider first of all at which points dπ¦ by dπ₯ is negative. Now, dπ¦ by dπ₯ 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 π΄, we can see that it does indeed slope downwards. So, dπ¦ by dπ₯ is negative at point π΄. At points π΅ and πΆ however, the tangents each slope upwards, which tells us that dπ¦ by dπ₯ will be positive at these two points. At point π·, the tangent appears to be horizontal, which means that dπ¦ by dπ₯ will be equal to zero at point π·, not negative. Finally, at point πΈ, we sketch in the tangent. And we see that it is indeed sloping downwards. So, dπ¦ by dπ₯ is negative at point πΈ. Weβre, therefore, left with only two possibilities, point π΄ and point πΈ.

The second condition is that d two π¦ by dπ₯ squared, thatβs the second derivative of our function π¦ with respect to π₯, 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 dπ¦ by dπ₯ 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 dπ¦ by dπ₯ is increasing, then this means that its derivative d two π¦ by dπ₯ 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 weβre 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, weβve narrowed our options down to points π΄ and πΈ. From the figure, we can see the at point π΄, the tangent is indeed below the curve. And therefore, the curve is concave upward at point π΄. And so, the second derivative is positive. However, at point πΈ, the tangent is above the curve. And so, the curve is concave downward at point πΈ, 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 dπ¦ by dπ₯ is negative, but d two π¦ by dπ₯ squared is positive is point π΄.