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ou must state the required derivative(s).
275. Find and identify any stationary points for
f(x)=e^(x)-4x+2.
For what values of
x is the function concave up?

ou must state the required derivative(s). $\newline$ $275$ . Find and identify any stationary points for $f(x)=e^{x}-4 x+2$ . $\newline$ For what values of $x$ is the function concave up?

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Solve trigonometric equations I

Full solution

Q. ou must state the required derivative(s).

\newline

275

. Find and identify any stationary points for

f(x)=e^{x}-4 x+2

.

\newline

For what values of

x

is the function concave up?

Find Derivative of $f(x)$ : First, let's find the derivative of $f(x)$ to locate stationary points. $\newline$ $f'(x) = \frac{d}{dx} [e^{x} - 4x + 2]$ $\newline$ $f'(x) = e^{x} - 4$
Locate Stationary Points: Now, set the derivative equal to zero to find stationary points. $e^{x} - 4 = 0$ $e^{x} = 4$
Solve for x: Solve for x. $\newline$ $x = \ln(4)$
Find Second Derivative: To identify the nature of the stationary point, find the second derivative. $\newline$ $f''(x) = \frac{d}{dx} [e^{x} - 4]$ $\newline$ $f''(x) = e^{x}$
Check Concavity: Plug in the value of $x$ into the second derivative to check concavity. $f''(\ln(4)) = e^{\ln(4)}$ $f''(\ln(4)) = 4$
Identify Minimum Point: Since $f''(\ln(4)) > 0$ , the function is concave up at $x = \ln(4)$ , and this is a minimum point.
Find Concave Up Points: To find where the function is concave up, look for where $f''(x) > 0$ . Since $f''(x) = e^{x}$ and $e^{x}$ is always positive, the function is concave up for all $x$ .

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