If a continuous function ƒ defined on the real line R, assumes positive and negative values in R, then the equation f(x) = 0 has a root in R,

Question

Read the following passage and answer the questions.

If a continuous function ƒ defined on the real line R, assumes positive and negative values in R, then the equation f(x) = 0 has a root in R, for example, if it is known that a continous function f on R is positive at some point and its minimum value is negative , then the equations f(x) = 0 has a root in R. Consider f(x) = ke^x –x, ∀ x ∈ R where k ∈ R is a constant.

(i) The line y = x meets y = ke^x for k ≤ 0 at

(a) no point

(b) one point

(c) two point

(d) more than two points

(ii) The value of k for which ke^x – x = 0 has only one root is

(a) 1/e

(b) e

(c) log_e2

(d) 1

(iii) For k > 0, the set of all values of k for which ke^x – x = 0 has two distinct roots is

(a) \((0, \frac 1e)\)

(b) \((\frac1 e,1)\)

(c) \((\frac1 e,\infty)\)

(d) (0, 1)

Answer 1

(i) (b) one point

(ii) (a) 1/e

(iii) (a) \((0, \frac 1e)\)