Video Transcript
The graph of 𝑓 is shown in the figure and 𝑓 is twice differentiable. Which of the following has the smallest value? 1) 𝑓 of one. 2) 𝑓 prime of one. 3) 𝑓 double prime of one.
So we’ve been given a graph of this function 𝑓, which we’re told is twice differentiable, and asked to determine whether the function, the first derivative of the function, or the second derivative of the function have the smallest value when 𝑥 is equal to one. We can read the value of 𝑓 of one from the graph itself. We see that the graph of 𝑓 crosses the 𝑥-axis when 𝑥 is equal to one. And therefore, 𝑓 of one is equal to zero.
Now, let’s consider 𝑓 prime of one. And to do this, we recall that 𝑓 prime of 𝑥 gives the slope of the curve at a particular point. We can draw in a tangent to the curve at 𝑥 equals one. And we see that this tangent slopes upwards. We don’t know the value of 𝑓 prime of one. But we can deduce that it is positive. So we now know that 𝑓 of one is equal to zero and 𝑓 prime of one is greater than zero. But what about 𝑓 double prime of one?
Well, by drawing in some more tangents in the vicinity of 𝑥 equals one, we note that the graph is concaved down at 𝑥 equals one because each of these tangents lie above the curve itself. If a function is concaved downwards, then its slope is decreasing. The slope of the function 𝑓 is 𝑓 prime of 𝑥. And now we recall that if a function is decreasing, then its slope is negative. The slope of 𝑓 prime of 𝑥 is its derivative. That’s 𝑓 double prime of 𝑥. So we can deduce that at 𝑥 equals one 𝑓 double prime of 𝑥 is less than zero.
We’ve now seen that 𝑓 of one is equal to zero, 𝑓 prime of one is greater than zero, and 𝑓 double prime of one is less than zero. The smallest value then will be 𝑓 double prime of one. And as it is strictly the smallest, we can conclude that of the three the one which has the smallest value is number three only.
