MathematicsAdvancedHigh School (11–12)Functions and Calculus › Application of DerivativesJEE Advanced
Created by StemPy LabA function is differentiable at every real number, and its derivative is continuous. The figure shows the graph of , the derivative, not itself. The graph of meets the -axis only at , and : it crosses the axis at and at , and only touches it at . Elsewhere is positive for and for , and negative in between (apart from , where it is zero). Let for every real . How many points of local extremum (local maximum or local minimum) does have?
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← Let f(x) = |x² - 1| for every real x. The figure shows the graph of f: a parabola whose part between x = -1 and x = 1 has been reflected above the x-axis, so that f has corners at x = -1 and x = 1 and a local maximum at x = 0. Now let h(x) = f(f(x)), the function f applied twice. A student argues that h must keep the two corners of f, and must also gain new corners wherever f(x) = 1. At how many real numbers x is h not differentiable?Consider the equation x⁶ = 6ˣ, where x is a real number. It is easy to see that x = 6 is a solution, since both sides are then 6⁶. A class is asked whether there are any others. One student says that a power of x grows more slowly than an exponential, so the two sides can be equal only once. Another says that a negative x cannot work because 6ˣ is then less than 1. Count every real solution, whether positive, zero or negative. You may use 2 < e < 3. How many real numbers x satisfy x⁶ = 6ˣ? →
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