The function 𝑓 is defined by 𝑓(𝑥)=𝑥𝑥+4 𝑓(𝑥)=cos𝑥. What point(s) (𝑥,𝑦) on the graph of 𝑓 have the property that the line tangent to 𝑓 at (𝑥,𝑦) has slope 14 −22?
Question
The function 𝑓 is defined by 𝑓(𝑥)=𝑥𝑥+4 𝑓(𝑥)=cos𝑥. What point(s) (𝑥,𝑦) on the graph of 𝑓 have the property that the line tangent to 𝑓 at (𝑥,𝑦) has slope 14 −22?
Solution
The slope of the tangent line to the graph of a function at a given point is given by the derivative of the function at that point. Therefore, we need to find the value(s) of x for which the derivative of f(x) is equal to 14 - 22 = -8.
First, let's find the derivative of the function f(x) = cos(x). Using the chain rule, we get:
f'(x) = -sin(x)
We set this equal to -8 and solve for x:
-sin(x) = -8
This equation has no solution, because the sine function outputs values between -1 and 1, and -8 is not in this interval.
Therefore, there are no points (x, y) on the graph of f(x) = cos(x) where the slope of the tangent line is -8.
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