Mathematics

Mathematics

Problem

Consider the function f(x)=−2x2+8x+3f(x)=-2x^{2}+8x+3 on the closed interval [0,5][0,5]. At which xx–value does ff attain its absolute maximum on this interval? a) x=4x=4 b) x=0x=0 c) ff does not attain an absolute maximum on [0,5][0,5] d) x=2x=2
Intuitive explanation

The key is recognizing that a continuous function on a closed interval reaches its extreme values either at a critical point (where the derivative is zero) or at the boundary endpoints. Find where f′(x)=0f'(x) = 0 to locate the critical point, then evaluate ff at that point and at both x=0x = 0 and x=5x = 5 to compare which gives the largest value.

Intuitive explanation

The key is recognizing that a continuous function on a closed interval reaches its extreme values either at a critical point (where the derivative is zero) or at the boundary endpoints. Find where f′(x)=0f'(x) = 0 to locate the critical point, then evaluate ff at that point and at both x=0x = 0 and x=5x = 5 to compare which gives the largest value.

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