48. multiple choice if f is a continuous, decreasing function on 0, 10 with a critical point at (4, 2)…

48. multiple choice if f is a continuous, decreasing function on 0, 10 with a critical point at (4, 2), which of the following statements must be false? (a) f(10) is an absolute minimum of f on 0, 10. (b) f(4) is neither a relative maximum nor a relative minimum. (c) f(4) does not exist. (d) f(4)=0 (e) f(4)<0 49. multiple choice which of the following functions has exactly two local extrema on its domain? (a) f(x)=|x - 2| (b) f(x)=x^3 - 6x + 5 (c) f(x)=x^3 + 6x - 5 (d) f(x)=tan x (e) f(x)=x + ln x 50. multiple choice if an even function f with domain all real numbers has a local maximum at x = a, then f(-a) (a) is a local minimum. (b) is a local maximum. (c) is both a local minimum and a local maximum. (d) could be either a local minimum or a local maximum. (e) is neither a local minimum nor a local maximum.

48. multiple choice if f is a continuous, decreasing function on 0, 10 with a critical point at (4, 2), which of the following statements must be false? (a) f(10) is an absolute minimum of f on 0, 10. (b) f(4) is neither a relative maximum nor a relative minimum. (c) f(4) does not exist. (d) f(4)=0 (e) f(4)<0 49. multiple choice which of the following functions has exactly two local extrema on its domain? (a) f(x)=|x - 2| (b) f(x)=x^3 - 6x + 5 (c) f(x)=x^3 + 6x - 5 (d) f(x)=tan x (e) f(x)=x + ln x 50. multiple choice if an even function f with domain all real numbers has a local maximum at x = a, then f(-a) (a) is a local minimum. (b) is a local maximum. (c) is both a local minimum and a local maximum. (d) could be either a local minimum or a local maximum. (e) is neither a local minimum nor a local maximum.

Answer

Explanation:

Step1: Analyze question 48

A decreasing function on [0, 10] with a critical - point at (4, 2). Since it's decreasing, the derivative at any point in the domain where it's differentiable is negative. A critical point can be where the derivative is 0 or does not exist, but for a decreasing function, if it's differentiable at the critical point, the derivative is 0.

Step2: Evaluate each option for question 48

  • Option A: Since (f) is decreasing on [0, 10], (f(10)) is the absolute minimum. This is true.
  • Option B: A critical point of a decreasing function is not a relative extremum. This is true.
  • Option C: A critical point can be where the derivative does not exist. This is possible.
  • Option D: A critical point can be where the derivative is 0. This is possible.
  • Option E: Since (f) is decreasing, if (f) is differentiable at (x = 4), (f^{\prime}(4)<0) is false because at a critical - point, if differentiable, (f^{\prime}(4) = 0).

Step3: Analyze question 49

Find the derivatives of each function and determine the number of local extrema.

  • For (y = f(x)=\vert x - 2\vert), (y=\begin{cases}x - 2, & x\geq2\-(x - 2), & x<2\end{cases}), it has one local minimum at (x = 2).
  • For (y=f(x)=x^{3}-6x + 5), (y^{\prime}=3x^{2}-6=3(x^{2}-2)=3(x-\sqrt{2})(x + \sqrt{2})). There are two local extrema (a local maximum at (x=-\sqrt{2}) and a local minimum at (x = \sqrt{2})).
  • For (y=f(x)=x^{3}+6x - 5), (y^{\prime}=3x^{2}+6>0) for all (x), so no local extrema.
  • For (y = f(x)=\tan x), it has infinitely many local extrema.
  • For (y=f(x)=x+\ln x), (y^{\prime}=1+\frac{1}{x}), (x>0), (y^{\prime}>0) for all (x>0), so no local extrema.

Step4: Analyze question 50

An even function satisfies (f(x)=f(-x)) for all (x) in the domain. If (f(x)) has a local maximum at (x = a), then (f(a)\geq f(x)) for (x) in some open interval around (a). Since (f(-x)=f(x)), (f(-a)=f(a)) and (f(-a)\geq f(-x)) for (-x) in some open interval around (-a). So (f(-a)) is also a local maximum.

Answer:

  1. E. (f^{\prime}(4)<0)
  2. B. (f(x)=x^{3}-6x + 5)
  3. B. is a local maximum.