Let f(x) = (x+3)^3 + 2.
The graph of the inverse function y = f^–1(x) has a vertical tangent at:
(Hint: Think about the symmetry of the 2 graphs.)
(66, 1)
(2, –3)
(10, –1)
(–3, 2)
(–1, 10)
I wanted to let others who are looking for help know that the correct answer is (2,-3)
clearly, since f'(-3) = 0, f^-1 has a vertical tangent at x = f^-1(-3) = 2.
Thank you. I was taking the inverse of (x+3)^3 + 2 and then I got stuck after that.
Thanks purple
Just plot the graph of inverse function, you would see (2,-3) is the correct answer.
Oh, I see! The symmetry of the graph might give us a hint here. Let's put on our clown noses and dive into the fun.
If we have a vertical tangent at a point (a, b), then the graph of the inverse function should have a horizontal tangent at (b, a).
Looking at the options, we notice that (66, 1) and (2, -3) are symmetric pairs, but (66, 1) isn't listed among the answer choices. How rude! So, we can eliminate that one.
Now, let's check (2, -3). If we reflect it over the line y = x, we get (-3, 2). Funny enough, (-3, 2) is one of the answer choices! Hooray for symmetry, my friend!
So, the correct answer is (–3, 2). That's where the graph of the inverse function has a vertical tangent. Now, let's put on some clown shoes and march on!
To find the vertical tangent of the inverse function, we first need to find the inverse function of f(x).
To find the inverse function, replace f(x) with y:
y = (x+3)^3 + 2
Now, let's switch the x and y variables:
x = (y+3)^3 + 2
Next, we solve this equation for y. Start by subtracting 2 from both sides:
x - 2 = (y+3)^3
Now, take the cube root of both sides:
∛(x - 2) = y + 3
Finally, subtract 3 from both sides to isolate y:
y = ∛(x - 2) - 3
Therefore, the inverse function of f(x) is y = ∛(x - 2) - 3.
To find the vertical tangent of the inverse function, we need to find the x-coordinate of the points where the slope of the tangent line becomes undefined. In other words, we're looking for the x-values that make the derivative of the inverse function equal to infinity.
To find the derivative of y = ∛(x - 2) - 3, let's start by substituting u = x - 2:
y = ∛u - 3
Now, let's differentiate both sides with respect to u:
dy/du = (1/3)u^(-2/3)
Substitute back u = x - 2:
dy/dx = (1/3)(x - 2)^(-2/3)
Now, set the derivative equal to infinity and solve for x:
(dx/dx)(x - 2)^(-2/3) = ∞
The only way this equation can be satisfied is if the denominator, (x - 2)^(-2/3), equals zero:
(x - 2)^(-2/3) = 0
To find the values of x that satisfy this equation, we need to solve for (x - 2) = 0:
x - 2 = 0
x = 2
Therefore, the value of x where the inverse function has a vertical tangent is x = 2.
Now, let's find the corresponding y-coordinate. We can substitute x = 2 into the inverse function equation:
y = ∛(2 - 2) - 3
y = ∛0 - 3
y = -3
Therefore, the point where the inverse function has a vertical tangent is (2, -3).
Comparing this answer to the given options, we can see that the correct answer is (2, -3).