Algebraic Methods, If Any?

Algebra Level 3

e x ⩾ a x e^x \geqslant ax for all x ⩾ 0 x \geqslant 0 and some real number a a . What is the maximum possible value of a a ?

NOTE: Non-algebraic solution ahead.


The answer is 2.71828.

Arkajyoti Banerjee by Arkajyoti Banerjee , 20, India

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2 solutions

Using the equality as the limit: a = e x x a=\frac{e^x}{x} . Next, look for extrema, ∂ e x x ∂ x = 0 \frac{\partial \frac{e^x}{x}}{\partial x}=0 . ∂ e x x ∂ x ⟹ e x x − e x x 2 ⟹ x = 1 \frac{\partial \frac{e^x}{x}}{\partial x} \Longrightarrow \frac{e^x}{x}-\frac{e^x}{x^2} \Longrightarrow x=1 , Substituting that back into the original equation: a = e a=e .

1 Helpful 1 Interesting 1 Brilliant 0 Confused

I don't quite know how to approach this problem using algebra, but a graphical method worked (WITHOUT USING GRAPHING CALCULATORS, OBVIOUSLY!).

It is quite obvious that the largest value of a a is clearly the slope of the line tangent to y = e x y = e^x passing through the origin.

Suppose the line is tangent to y = e x y = e^x at ( x 1 , a x 1 ) (x_1, ax_1) , we then also have a x 1 = e x 1 ax_1 = e^{x_1} .

Slope of tangent to y = e x y = e^x at ( x 1 , a x 1 ) = e x 1 (x_1, ax_1) = e^{x_1}

Equation of the line : y − a x 1 x − x 1 = e x 1 \frac{y-ax_1}{x-x_1} = e^{x_1}

Since this line passes through the origin,

0 − a x 1 0 − x 1 = e x 1 ⟹ a = e x 1 \frac{0-ax_1}{0-x_1} = e^{x_1} \implies a = e^{x_1} .

Substituting this in a x 1 = e x 1 ax_1 = e^{x_1} , we have x 1 = 1 x_1 = 1 .

Therefore, a = e 1 = e a = e^1 = e .

Hence, the maximum value of a a is e e .

1 Helpful 1 Interesting 0 Brilliant 0 Confused

Pretty algebraic solution if you asked me. Lot of words that weren't necessary :p. You simply used the fact that: f ′ ( x ) = f ( x ) − y a x − x a f'(x) = \frac{f(x) - y_a} {x - x_a} with ( x a , y a ) (x_a, y_a) being the known point on the tangent line(in this case the origin).

Peter van der Linden - 3 years, 2 months ago

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Well, wanted to write a 'detailed' solution after many days.

I used a method which was graphical in the making. Coming to the conclusion that "the maximum value of a a is the slope of the line tangent to the curve passing through the origin.", this is more geometry-centred, isn't it?

Arkajyoti Banerjee - 3 years, 2 months ago

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kut-kut,aeeeeeeeeeeeeee dooooodoooooooooooo

Aman Banka - 3 years, 2 months ago

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