1 Advanced Mathematics • Real Analysis Let f:[0,infinity) be such that df/dx is monotonically increasing and f(0) = 0. If g(x) = f(x)/x, then g is: Let f:[0,infinity) be such that df/dx is monotonically increasing and f(0) = 0. If g(x) = f(x)/x, then g is: सेव करें Save सेव्ड Saved A. Concave function Concave function B. Convex function Convex function C. Monotonically decreasing Monotonically decreasing D. Monotonically increasing Monotonically increasing उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: D Correct Answer: D An increasing derivative makes f convex. For a convex function with f(0)=0, the secant slope f(x)/x from the origin is monotonically increasing. An increasing derivative makes f convex. For a convex function with f(0)=0, the secant slope f(x)/x from the origin is monotonically increasing.
2 Advanced Mathematics • Real Analysis The function sin(e^x), where x > 0: The function sin(e^x), where x > 0: सेव करें Save सेव्ड Saved A. Is bounded but does not converge Is bounded but does not converge B. Diverges to infinity as x approaches infinity Diverges to infinity as x approaches infinity C. Is strictly increasing but bounded Is strictly increasing but bounded D. Is bounded and converges to a real number as x approaches infinity Is bounded and converges to a real number as x approaches infinity उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: A Correct Answer: A The sine function always lies between -1 and 1, but e^x grows without bound and its sine continues to oscillate. Thus the function is bounded but has no limit. The sine function always lies between -1 and 1, but e^x grows without bound and its sine continues to oscillate. Thus the function is bounded but has no limit.
3 Advanced Mathematics • Real Analysis Let f(x) be a differentiable function on R such that df/dx > 0 for all x and lim f(x) = 2 as x approaches minus infinity. Which of the following is true? Let f(x) be a differentiable function on R such that df/dx > 0 for all x and lim f(x) = 2 as x approaches minus infinity. Which of the following is true? सेव करें Save सेव्ड Saved A. f(3) > 2 f(3) > 2 B. f(3) >= 2 f(3) >= 2 C. f(3) <= 2 f(3) <= 2 D. f(3) = 2 f(3) = 2 उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: A Correct Answer: A Since f is strictly increasing and approaches 2 at minus infinity, every finite value of f must be strictly greater than 2. Therefore f(3) > 2. Since f is strictly increasing and approaches 2 at minus infinity, every finite value of f must be strictly greater than 2. Therefore f(3) > 2.
4 Advanced Mathematics • Real Analysis Function f(x)=|x|+sin(x) is: Function f(x)=|x|+sin(x) is: सेव करें Save सेव्ड Saved A. Convex Convex B. Continuous and differentiable at x=0 Continuous and differentiable at x=0 C. Continuous but not differentiable at x=0 Continuous but not differentiable at x=0 D. Concave Concave उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: C Correct Answer: C Both terms are continuous. At zero, the one-sided derivatives are -1+1=0 and 1+1=2, so the derivative does not exist. Both terms are continuous. At zero, the one-sided derivatives are -1+1=0 and 1+1=2, so the derivative does not exist.
5 Advanced Mathematics • Real Analysis Let f(x)=1/x^2-5 be defined on [1,4]. Which of the following is true? Let f(x)=1/x^2-5 be defined on [1,4]. Which of the following is true? सेव करें Save सेव्ड Saved A. There exists a point x in [1,4] such that f is not differentiable at x. There exists a point x in [1,4] such that f is not differentiable at x. B. The function has a tangent line between 1 and 4 with slope -1/64. The function has a tangent line between 1 and 4 with slope -1/64. C. The function has a tangent line between 1 and 4 with slope 3/2. The function has a tangent line between 1 and 4 with slope 3/2. D. The function has a tangent line between 1 and 4 with slope -5/16. The function has a tangent line between 1 and 4 with slope -5/16. उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: D Correct Answer: D The secant slope is [f(4)-f(1)]/(4-1)=[1/16-1]/3=-5/16. By the Mean Value Theorem, some tangent has this slope. The secant slope is [f(4)-f(1)]/(4-1)=[1/16-1]/3=-5/16. By the Mean Value Theorem, some tangent has this slope.