Question Video: Identifying the Concavity and Monotonicity of a Function from Its Graph | Nagwa Question Video: Identifying the Concavity and Monotonicity of a Function from Its Graph | Nagwa

Question Video: Identifying the Concavity and Monotonicity of a Function from Its Graph Mathematics • Third Year of Secondary School

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The graph of a function ๐‘ฆ = ๐‘“(๐‘ฅ) is shown, at which point are d๐‘ฆ/d๐‘ฅ and dยฒ๐‘ฆ/d๐‘ฅยฒ both negative?

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Video Transcript

The graph of a function ๐‘ฆ equals ๐‘“ of ๐‘ฅ is shown, at which point are d๐‘ฆ by d๐‘ฅ and d two ๐‘ฆ by d๐‘ฅ squared both negative.

So weโ€™ve been given the graph of a function itself. And weโ€™re asked to use this to determine at which of these five points both the first and second derivatives of the function are negative. First, letโ€™s consider the sign of the first derivative d๐‘ฆ by d๐‘ฅ at each point. Recall that the first derivative of a function at a point gives the slope of the tangent to the curve at that point. So by sketching in tangents to the curve at each point, we can determine the sign of their first derivative.

We see that at point ๐ด the tangent is sloping downwards. So the first derivative, d๐‘ฆ by d๐‘ฅ, is indeed negative at point ๐ด. However, at points ๐ต and ๐ถ, the tangents are each sloping upwards, which tells us that the first derivative d๐‘ฆ by d๐‘ฅ will be positive at both ๐ต and ๐ถ. At point ๐ท, the tangent to the curve is horizontal. So the first derivative will be equal to zero, not negative at this point. Finally, at point ๐ธ, we see that the tangent is sloping downwards. So the first derivative will also be negative at point ๐ธ. Weโ€™re, therefore, left with only two options ๐ด and ๐ธ. Next, we need to consider the sign of the second derivative at each of these points. And this is linked to the concavity of the curve at each point.

Recall that the curve is said to be concave downward on a particular interval, if the tangents to the curve in that interval lie above the curve itself. We can also see that when a curve is concave downwards, the slope of its tangent is decreasing. And therefore, its first derivative is also decreasing. When a function is decreasing, then its derivative is negative. And as the derivative of the first derivative is the second derivative, it follows that d two ๐‘ฆ by d๐‘ฅ squared will be less than zero when a curve is concave downward.

This isnโ€™t required here, but a curve is said to be concave upward when the reverse is true. The tangents to the graph lie below the graph itself. The first derivative is increasing and therefore, the second derivative is positive. By considering the graph of ๐‘“ of ๐‘ฅ, we can see that the tangent we drew at point ๐ธ lies above the curve. And indeed, the shape of the curve is concave downward in this region. However, if we look at point ๐ด, the tangent we drew here is below the curve. And so the graph is concave upward at point ๐ด. This tells us then the second derivative will be negative at point ๐ธ, whereas it will be positive at point ๐ด.

Weโ€™re left then with only one point at which both the first and second derivatives are negative. Itโ€™s point ๐ธ. Weโ€™ve seen in this example how to determine something about the first and second derivatives of a function from a graph of the function itself.

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