The Maximized Slope

Calculus Level 5

Let function $f$ be defined as $f(x)=(\cos(a))^x+(\sin(a))^x$ where $a$ is a parameter that is constantly changing within the interval $0<a<\frac{\pi}{2}$ .

The $y$ -intercept's tangent line is maximized when the value of $a$ reaches a certain value.

If the value of the maximized slope is expressed as $\ln($ $\frac{m}{n}$ $)$ where $m$ and $n$ are coprime integers, find the value of $m+n$ .

The answer is 3.

1 solution

Oli Hohman
Apr 22, 2016

f(x) = (cos(a))^x+(sin(a))^x f'(x) = sin(a)^x ln(sin(a))+cos(a)^x ln(cos(a))

The problem says that the y-intercept's tangent line is maximized when a reaches a certain value. The y-intercept is the equation x=0, so you find f'(0) in order to find the a for which f'(a) is maximized, then calculate f'(a) and express it appropriately.

f'(0) = ln(sin(a))+ln(cos(a)) (f'(0)) differentiated with respect to a yields cot(a)-tan(a) = 0 sin^2(a)=cos^2(a) a=pi/4 Plug in a=pi/4 into f'(0)

ln(sin(pi/4))+ln(cos(pi/4)) = ln(1/sqrt(2))+ln(1/sqrt(2)) = ln(1/2) by properties of logs.

Therefore m + n = 1 + 2 = 3

The Maximized Slope

The answer is 3.

1 solution

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