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anonymous

  • 4 years ago

4. Let f(x) = (1/pi^3)x^3 + (9/pi)x + sinx a) Show that f'(x) is never 0. Attempt: f'(x) = (3/pi^3)x^2 + (9/pi) + cosx cos(x)'s highest y-val's are -1, 1, and (3/pi^3)x^2 + (9/pi) > 0 always Even if, f'(0) = 0 + (9/pi) + 1 > 0 So from examining cosine's range restrictions and the presence of a squared term, I can infer that f'(x) can never be 0. Is this a fair way of showing this? b) Explain why f must be invertible Not sure how I am going to go about with this one. I tried to swap x values and try to isolate for x, then interchange them to make the inverse function.....

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  1. anonymous
    • 4 years ago
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    4. Let f(x) = (1/pi^3)x^3 + (9/pi)x + sinx a) Show that f'(x) is never 0. Attempt: f'(x) = (3/pi^3)x^2 + (9/pi) + cosx cos(x)'s highest y-val's are -1, 1, and (3/pi^3)x^2 + (9/pi) > 0 always Even if, f'(0) = 0 + (9/pi) + 1 > 0 So from examining cosine's range restrictions and the presence of a squared term, I can infer that f'(x) can never be 0. Is this a fair way of showing this? b) Explain why f must be invertible Not sure how I am going to go about with this one. I tried to swap x values and try to isolate for x, then interchange them to make the inverse function, but I quickly realized that I cannot use that method to get the inverse function. I definitely need a hint for this one. c) Show that the point (10, pi) is on the graph of the inverse fcn If (pi, 10) is on the graph of the function f and this point exists, then can I conclude from that that the point (10,pi) exists on the graph of the inverse fcn? d) Find the equation of the line tangent to the inverse fcn at (10, pi) My plan was to do this: a) Take the derivative of f b) Find the slope of f at x = pi c) Infer that because f(y) = x <=> f-1(x) = y, then the slope I get for this point on the function f will apply for the slope of the inverse fcn at x = 10. Is this even true?

  2. anonymous
    • 4 years ago
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    See the second post please. NOT FIRST

  3. anonymous
    • 4 years ago
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    you can find the vertex of the quadratic and show that it is above 1

  4. anonymous
    • 4 years ago
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    makes the function invertible because if the derivative is always positive the function is always increasing hence one to one

  5. anonymous
    • 4 years ago
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    don't expect to actually find the inverse as a function

  6. anonymous
    • 4 years ago
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    to show that \[(1,\pi)\] is on the graph of the inverse, show that \[f(\pi)=10\]

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