Find the value of x for which the function f(x) = e^x - x^2 has a horizontal tan

Find the value of x for which the function f(x) = e^x - x^2 has a horizontal tangent.

Find the value of x for which the function f(x) = e^x - x^2 has a horizontal tangent.

Step 1: Understand that a horizontal tangent means the slope of the function is zero.
Step 2: Find the derivative of the function f(x) = e^x - x^2. The derivative is f'(x) = e^x - 2x.
Step 3: Set the derivative equal to zero to find where the slope is zero: e^x - 2x = 0.
Step 4: Solve the equation e^x = 2x. This may require numerical methods or graphing to find the approximate solution.
Step 5: After solving, you find that x is approximately equal to 1.

Derivative and Tangent – The question tests the understanding of derivatives and the condition for a horizontal tangent, which occurs when the derivative equals zero.
Exponential and Polynomial Functions – The function involves both exponential and polynomial components, requiring knowledge of their behavior and interaction.