1 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: 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 Mathematics • Real Analysis The function sin(e^x), where x > 0: The function sin(e^x), where x > 0: 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 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? 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 Mathematics • Real Analysis Function f(x)=|x|+sin(x) is: Function f(x)=|x|+sin(x) is: 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 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? 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.