Video Transcript
The graph of a function ๐ฆ equals ๐ of ๐ฅ is shown. At which point is d๐ฆ by d๐ฅ negative but d two ๐ฆ by d๐ฅ squared positive.
From the figure, we can see that there are five points, ๐ด, ๐ต, ๐ถ, ๐ท, and ๐ธ. And we need to determine at which of these five points these two facts are true. Letโs consider first of all at which points d๐ฆ by d๐ฅ is negative. Now, d๐ฆ by d๐ฅ 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 ๐ด, we can see that it does indeed slope downwards. So, d๐ฆ by d๐ฅ is negative at point ๐ด. At points ๐ต and ๐ถ however, the tangents each slope upwards, which tells us that d๐ฆ by d๐ฅ will be positive at these two points. At point ๐ท, the tangent appears to be horizontal, which means that d๐ฆ by d๐ฅ will be equal to zero at point ๐ท, not negative. Finally, at point ๐ธ, we sketch in the tangent. And we see that it is indeed sloping downwards. So, d๐ฆ by d๐ฅ is negative at point ๐ธ. Weโre, therefore, left with only two possibilities, point ๐ด and point ๐ธ.
The second condition is that d two ๐ฆ by d๐ฅ squared, thatโs the second derivative of our function ๐ฆ with respect to ๐ฅ, 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 d๐ฆ by d๐ฅ 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 d๐ฆ by d๐ฅ is increasing, then this means that its derivative d two ๐ฆ by d๐ฅ 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 weโre 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, weโve narrowed our options down to points ๐ด and ๐ธ. From the figure, we can see the at point ๐ด, the tangent is indeed below the curve. And therefore, the curve is concave upward at point ๐ด. And so, the second derivative is positive. However, at point ๐ธ, the tangent is above the curve. And so, the curve is concave downward at point ๐ธ, 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 d๐ฆ by d๐ฅ is negative, but d two ๐ฆ by d๐ฅ squared is positive is point ๐ด.
